From ecbc8511a11ad347c5dcb327e1879ac3ba1768f7 Mon Sep 17 00:00:00 2001 From: sebapersson Date: Wed, 18 Feb 2026 09:38:27 +0000 Subject: [PATCH 1/6] Add first draft training approaches --- doc/index.rst | 6 ++ doc/training_approaches.rst | 206 ++++++++++++++++++++++++++++++++++++ 2 files changed, 212 insertions(+) create mode 100644 doc/training_approaches.rst diff --git a/doc/index.rst b/doc/index.rst index 35d0d90..b435a11 100644 --- a/doc/index.rst +++ b/doc/index.rst @@ -28,6 +28,12 @@ PEtab SciML - Scientific Machine Learning Format and Tooling layers tutorial +.. toctree:: + :caption: Tool developer guide + :maxdepth: 3 + + training_approaches + .. toctree:: :caption: Python package :maxdepth: 3 diff --git a/doc/training_approaches.rst b/doc/training_approaches.rst new file mode 100644 index 0000000..0fb42d3 --- /dev/null +++ b/doc/training_approaches.rst @@ -0,0 +1,206 @@ +Training strategies at the PEtab level +====================================== + +Training (parameter estimating) SciML models can be challenging, and often +standard ML training workflows (e.g., training with Adam for a fixed number of +epochs) fail to find a good minimum or require many training epochs. + +Several training strategies have been developed to address this. These include +curriculum learning, multiple shooting, and combined curriculum multiple +shooting, all of which can be implemented at the PEtab abstraction level for +ODE models as well as hybrid SciML problems. This page describes these +PEtab-level abstractions to allow tool developers to implement them. The PEtab +SciML library also provides a reference implementation. + +Curriculum learning +------------------- + +Curriculum learning is a training strategy where the optimization problem is +made progressively harder over successive curriculum stages. For PEtab +problems, a curriculum can be defined by gradually increasing the number of +measurement time points (and typically the simulation end time) over a fixed +number of stages. This can be implemented at the PEtab level as follows: + +Inputs: + +- A PEtab problem (PEtab v1 or v2). +- The number of curriculum stages, ``nStages``. +- A schedule ``n_i`` specifying how many measurements are included in stage + ``i``. + +1. Sort the measurement table in the input PEtab problem by the ``time`` + column. +2. Create ``nStages`` PEtab sub-problems. For stage ``i``, copy the original + PEtab problem and filter the time-sorted measurement table to keep the + first ``n_i`` measurements. +3. Optionally filter the condition and experiment tables to only include + entries required by the filtered measurement table for each sub-problem. + +A practical consideration for tools implementing and/or importing curriculum +problems is to keep parameter ordering consistent across stages, which +simplifies transferring parameter values between stages. + +.. _multiple_shooting: + +Multiple shooting +----------------- + +In multiple shooting, the simulation time span of each PEtab experiment is +split into windows that are fitted jointly. Each window has its own estimated +initial state values, and a continuity penalty is introduced to encourage a +continuous trajectory between adjacent windows. This can be implemented at the +PEtab level as follows: + +Inputs: + +- A PEtab problem (PEtab v2). +- The number of multiple-shooting windows, ``nWindows``. +- A window partition ``[t0_i, tf_i]`` for each window ``i = 1..nWindows`` such + that the union of windows covers the full measurement time range, and + ``t0_i != tf_i`` for all windows. +- A continuity penalty parameter ``lambda``. + +1. Copy the input PEtab problem to create a multiple shooting (MS) PEtab + problem. +2. In the MS PEtab problem, add the penalty weight parameter ``lambda`` to the + parameter table as a non-estimated parameter and set an appropriate nominal + value. +3. For each PEtab experiment with ID ``expId`` in the MS PEtab problem: + 1. Create ``nWindows`` new PEtab experiments with IDs ``WINDOW{i}_{expId}`` + and set the initial time to ``t0_i`` for window ``i = 1..nWindows``. + 2. In the experiment table, remove the original experiment IDs and keep + only the windowed experiments. Assign each PEtab condition to the + corresponding window experiment(s). If a PEtab condition occurs at a + time point that lies in the overlap of windows ``i-1`` and ``i``, assign + the condition to experiment ``WINDOW{i-1}_{expId}``. + 3. In the measurement table, assign all measurements in the time interval + ``[t0_i, tf_i]`` for experiment ``expId`` to experiment + ``WINDOW{i}_{expId}``. If MS windows overlap at time points that contain + measurements, duplicate those measurements so they appear in each + relevant window. + 4. For each window ``i > 1`` such that there exists at least one + measurement for ``expId`` at time ``t >= t0_i`` in the original problem + (i.e., at least one subsequent window contains measurements), assign + initial window values and a continuity penalty: + 1. In the parameter table, create parameters + ``WINDOW{i}_{expId}_init_stateId{j}`` for each model state + ``stateId{j}``. Mark them as estimated and choose appropriate bounds. + 2. In the condition table, create a condition with ID + ``WINDOW{i}_{expId}_condition0`` that assigns each ``stateId{j}`` to + ``WINDOW{i}_{expId}_init_stateId{j}``. + 3. Assign condition ``WINDOW{i}_{expId}_condition0`` as the initial + condition for experiment ``WINDOW{i}_{expId}`` at time ``t0_i``. + 4. In the observable table, create an observable with ID + ``WINDOW{i}_{expId}_penalty_stateId{j}`` for each model state + ``stateId{j}`` and set + + - ``observableFormula = sqrt(lambda) * (stateId{j} - WINDOW{i}_{expId}_init_stateId{j})`` + - ``noiseFormula = 1.0`` + - ``noiseDistribution = normal`` + + 5. In the measurement table, add a row for experiment + ``WINDOW{i}_{expId}`` and observable + ``WINDOW{i}_{expId}_penalty_stateId{j}`` at time ``t0_i`` with + ``measurement = 0.0``. This yields an L2 (quadratic) penalty. + +Naive multiple shooting can perform poorly when states have different scales, +since a single penalty weight may be impossible to tune. In this case, a +log-scale penalty such as + +``sqrt(lambda) * (log(stateId{j}) - log(WINDOW{i}_{expId}_init_stateId{j}))`` + +can be effective. + +From a runtime performance perspective, the number of initial-window +parameters scales with the number of windows, states, and PEtab experiments, +which can be impractical for larger problems. Finally, since initial-window +parameters must be estimated, this approach typically performs poorly for +partially observed systems; this is addressed by the curriculum multiple +shooting approach. + +Curriculum multiple shooting +---------------------------- + +Curriculum multiple shooting (CL+MS) combines multiple shooting with a +curriculum schedule. The idea is to start from a multiple-shooting formulation, +which is often easier to train, and then progressively reduce the number of +windows until the original (single-window) problem is recovered. This makes the +approach less sensitive to continuity-penalty tuning and ensures the final +parameters optimize the objective of the original PEtab problem. + +Practically, CL+MS defines ``nStages`` curriculum stages. Stage 1 corresponds +to a multiple-shooting problem with ``nWindows = nStages`` windows. In each +subsequent stage, the first ``nWindows-1`` windows are expanded to cover the +union of two adjacent windows, and the last window is dropped. This reduces +the number of windows by one per stage while increasing the time span covered +by each remaining window. The final stage has a single window and corresponds +to the original problem. This can be implemented at the PEtab level as follows: + +Inputs: + +- A PEtab problem (PEtab v2). +- The number of curriculum stages, ``nStages``. +- An initial window partition ``[t0_i, tf_i]`` for stage 1 with + ``i = 1..nStages``, such that the union of windows covers the full + measurement time range and ``t0_i != tf_i`` for all windows. +- A continuity penalty parameter ``lambda`` (used in the multiple-shooting + stages). + +1. Construct stage 1 as a multiple-shooting (MS) PEtab problem with + ``nWindows = nStages`` using the procedure in + :ref:`Multiple shooting `. +2. For curriculum stage ``k = 2..(nStages-1)``: + 1. Set the number of windows to ``nWindows = nStages - k + 1``. + 2. Define the MS window time spans for stage ``k`` by merging adjacent + windows from the previous stage: + - For ``i = 1..nWindows`` set ``t0_i^{(k)} = t0_i^{(k-1)}`` and + ``tf_i^{(k)} = tf_{i+1}^{(k-1)}``. + - Drop the last window of stage ``k-1``. + 3. Create the PEtab problem for stage ``k`` by applying the + :ref:`Multiple shooting ` construction with the + updated window partition. In particular: + + - Update the experiment table to contain only experiments + ``WINDOW{i}_{expId}`` for ``i = 1..nWindows``. + - Reassign and/or duplicate measurements to match + ``[t0_i^{(k)}, tf_i^{(k)}]``. + Measurements that in the original problem now appear in multiple + windows must be duplicated so they appear in each window. + - Include window-initial parameters and continuity-penalty observables + for windows ``i > 1`` as in multiple shooting. Note that the penalty + is applied at the initial time point of each window; in PEtab it is + not possible to define a continuity penalty over the full overlap + interval between two windows. + +3. The final stage (``nWindows = 1``) corresponds to the original PEtab + problem. Use the parameter estimate from stage ``nStages-1`` to initialize + optimization for the final stage. + +A practical consideration for tools implementing and/or importing CL+MS is that +the number of window-initial parameters to estimate changes between stages. To +support transferring parameter values between stages, it can be beneficial to +provide a utility function for mapping parameters from one stage problem to +the next. + +Partitioning measurements and time windows +------------------------------------------ + +The training approaches above require either splitting measurements into +curriculum stages (curriculum learning) or partitioning the simulation time +span into windows (multiple shooting and curriculum multiple shooting). We +recommend that tools supporting these methods provide the splitting schemes +outlined below. + +For curriculum learning, the number of measurements per stage, ``n_i``, can be +chosen in two ways: (i) split by unique measurement time points and allocate +``n_i`` accordingly, or (ii) split by the total number of measurements, which +can be effective when there are few unique time points but many repeated +measurements. We recommend supporting both modes, as well as automatic +splitting (e.g., given ``nStages``, compute ``n_i``) and user-defined +schedules (e.g., explicit ``n_i`` per stage or a maximum time point per stage). + +For multiple shooting, window intervals ``[t0_i, tf_i]`` must be defined. We +recommend supporting automatic window construction (e.g., take ``nWindows`` as +input and allocate windows based on unique measurement time points) as well as +user-specified intervals. As a basic sanity check, tools should ensure that +each window contains at least one measurement time point. From a136b476ecf07701aa784c568072d512ba08247a Mon Sep 17 00:00:00 2001 From: sebapersson Date: Wed, 18 Feb 2026 11:47:34 +0000 Subject: [PATCH 2/6] Update training approaches --- doc/training_approaches.rst | 33 +++++++++++++++++---------------- 1 file changed, 17 insertions(+), 16 deletions(-) diff --git a/doc/training_approaches.rst b/doc/training_approaches.rst index 0fb42d3..8061f4e 100644 --- a/doc/training_approaches.rst +++ b/doc/training_approaches.rst @@ -8,9 +8,9 @@ epochs) fail to find a good minimum or require many training epochs. Several training strategies have been developed to address this. These include curriculum learning, multiple shooting, and combined curriculum multiple shooting, all of which can be implemented at the PEtab abstraction level for -ODE models as well as hybrid SciML problems. This page describes these -PEtab-level abstractions to allow tool developers to implement them. The PEtab -SciML library also provides a reference implementation. +ODE models as well as hybrid PEtab SciML problems. This page describes these +PEtab-level abstractions for tool developers. The PEtab SciML library also +provides reference implementations. Curriculum learning ------------------- @@ -30,15 +30,15 @@ Inputs: 1. Sort the measurement table in the input PEtab problem by the ``time`` column. -2. Create ``nStages`` PEtab sub-problems. For stage ``i``, copy the original - PEtab problem and filter the time-sorted measurement table to keep the - first ``n_i`` measurements. +2. Create ``nStages`` PEtab sub-problems by copying the input problem. For + stage ``i``, filter the time-sorted measurement table to keep the first + ``n_i`` measurements. 3. Optionally filter the condition and experiment tables to only include entries required by the filtered measurement table for each sub-problem. A practical consideration for tools implementing and/or importing curriculum problems is to keep parameter ordering consistent across stages, which -simplifies transferring parameter values between stages. +simplifies transferring parameters between stages. .. _multiple_shooting: @@ -70,7 +70,7 @@ Inputs: and set the initial time to ``t0_i`` for window ``i = 1..nWindows``. 2. In the experiment table, remove the original experiment IDs and keep only the windowed experiments. Assign each PEtab condition to the - corresponding window experiment(s). If a PEtab condition occurs at a + corresponding window experiment. If a PEtab condition occurs at a time point that lies in the overlap of windows ``i-1`` and ``i``, assign the condition to experiment ``WINDOW{i-1}_{expId}``. 3. In the measurement table, assign all measurements in the time interval @@ -107,13 +107,14 @@ Naive multiple shooting can perform poorly when states have different scales, since a single penalty weight may be impossible to tune. In this case, a log-scale penalty such as -``sqrt(lambda) * (log(stateId{j}) - log(WINDOW{i}_{expId}_init_stateId{j}))`` +``sqrt(lambda) * (log(abs(stateId{j})) - log(WINDOW{i}_{expId}_init_stateId{j}))`` -can be effective. +can be effective, where ``abs`` avoid potential problems with states going +below zero due to numerical errors. From a runtime performance perspective, the number of initial-window parameters scales with the number of windows, states, and PEtab experiments, -which can be impractical for larger problems. Finally, since initial-window +which can be impractical for larger problems. Moreover, since initial-window parameters must be estimated, this approach typically performs poorly for partially observed systems; this is addressed by the curriculum multiple shooting approach. @@ -179,13 +180,12 @@ Inputs: A practical consideration for tools implementing and/or importing CL+MS is that the number of window-initial parameters to estimate changes between stages. To support transferring parameter values between stages, it can be beneficial to -provide a utility function for mapping parameters from one stage problem to -the next. +provide a utility function for mapping parameters between stage problems. Partitioning measurements and time windows ------------------------------------------ -The training approaches above require either splitting measurements into +The above training approaches above require either splitting measurements into curriculum stages (curriculum learning) or partitioning the simulation time span into windows (multiple shooting and curriculum multiple shooting). We recommend that tools supporting these methods provide the splitting schemes @@ -196,8 +196,9 @@ chosen in two ways: (i) split by unique measurement time points and allocate ``n_i`` accordingly, or (ii) split by the total number of measurements, which can be effective when there are few unique time points but many repeated measurements. We recommend supporting both modes, as well as automatic -splitting (e.g., given ``nStages``, compute ``n_i``) and user-defined -schedules (e.g., explicit ``n_i`` per stage or a maximum time point per stage). +splitting (e.g., given ``nStages``, compute ``n_i`` for the user) and +user-defined schedules (e.g., explicit ``n_i`` per stage or a maximum time +point per stage). For multiple shooting, window intervals ``[t0_i, tf_i]`` must be defined. We recommend supporting automatic window construction (e.g., take ``nWindows`` as From f461445e8f4b8b3b1f836f50d3c121506614ec00 Mon Sep 17 00:00:00 2001 From: sebapersson Date: Thu, 19 Feb 2026 08:05:35 +0000 Subject: [PATCH 3/6] Improve description of training schemes --- doc/training_approaches.rst | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/doc/training_approaches.rst b/doc/training_approaches.rst index 8061f4e..eb6e673 100644 --- a/doc/training_approaches.rst +++ b/doc/training_approaches.rst @@ -1,5 +1,5 @@ -Training strategies at the PEtab level -====================================== +SciML Training strategies at the PEtab level +============================================ Training (parameter estimating) SciML models can be challenging, and often standard ML training workflows (e.g., training with Adam for a fixed number of @@ -15,7 +15,7 @@ provides reference implementations. Curriculum learning ------------------- -Curriculum learning is a training strategy where the optimization problem is +Curriculum learning is a training strategy where the training problem is made progressively harder over successive curriculum stages. For PEtab problems, a curriculum can be defined by gradually increasing the number of measurement time points (and typically the simulation end time) over a fixed @@ -33,8 +33,8 @@ Inputs: 2. Create ``nStages`` PEtab sub-problems by copying the input problem. For stage ``i``, filter the time-sorted measurement table to keep the first ``n_i`` measurements. -3. Optionally filter the condition and experiment tables to only include - entries required by the filtered measurement table for each sub-problem. +3. Optionally filter the condition, observable and experiment tables to only + include entries required by the measurement table for each sub-problem. A practical consideration for tools implementing and/or importing curriculum problems is to keep parameter ordering consistent across stages, which @@ -173,9 +173,9 @@ Inputs: not possible to define a continuity penalty over the full overlap interval between two windows. -3. The final stage (``nWindows = 1``) corresponds to the original PEtab - problem. Use the parameter estimate from stage ``nStages-1`` to initialize - optimization for the final stage. +3. The final stage corresponds to the original PEtab problem. Use the parameter + estimate from stage ``nStages-1`` to initialize optimization for the final + stage. A practical consideration for tools implementing and/or importing CL+MS is that the number of window-initial parameters to estimate changes between stages. To @@ -204,4 +204,4 @@ For multiple shooting, window intervals ``[t0_i, tf_i]`` must be defined. We recommend supporting automatic window construction (e.g., take ``nWindows`` as input and allocate windows based on unique measurement time points) as well as user-specified intervals. As a basic sanity check, tools should ensure that -each window contains at least one measurement time point. +each window contains at least one measurement. From 60a8f35f97ebc7b096856deb3102d9bb485c75e8 Mon Sep 17 00:00:00 2001 From: Sebastian Persson <46872750+sebapersson@users.noreply.github.com> Date: Fri, 20 Feb 2026 14:51:42 +0000 Subject: [PATCH 4/6] Apply suggestions from code review Co-authored-by: Dilan Pathirana <59329744+dilpath@users.noreply.github.com> Co-authored-by: BSnelling --- doc/training_approaches.rst | 9 +++++++-- 1 file changed, 7 insertions(+), 2 deletions(-) diff --git a/doc/training_approaches.rst b/doc/training_approaches.rst index eb6e673..6ecd670 100644 --- a/doc/training_approaches.rst +++ b/doc/training_approaches.rst @@ -66,6 +66,7 @@ Inputs: parameter table as a non-estimated parameter and set an appropriate nominal value. 3. For each PEtab experiment with ID ``expId`` in the MS PEtab problem: + 1. Create ``nWindows`` new PEtab experiments with IDs ``WINDOW{i}_{expId}`` and set the initial time to ``t0_i`` for window ``i = 1..nWindows``. 2. In the experiment table, remove the original experiment IDs and keep @@ -82,6 +83,7 @@ Inputs: measurement for ``expId`` at time ``t >= t0_i`` in the original problem (i.e., at least one subsequent window contains measurements), assign initial window values and a continuity penalty: + 1. In the parameter table, create parameters ``WINDOW{i}_{expId}_init_stateId{j}`` for each model state ``stateId{j}``. Mark them as estimated and choose appropriate bounds. @@ -151,12 +153,15 @@ Inputs: ``nWindows = nStages`` using the procedure in :ref:`Multiple shooting `. 2. For curriculum stage ``k = 2..(nStages-1)``: + 1. Set the number of windows to ``nWindows = nStages - k + 1``. 2. Define the MS window time spans for stage ``k`` by merging adjacent windows from the previous stage: + - For ``i = 1..nWindows`` set ``t0_i^{(k)} = t0_i^{(k-1)}`` and ``tf_i^{(k)} = tf_{i+1}^{(k-1)}``. - Drop the last window of stage ``k-1``. + 3. Create the PEtab problem for stage ``k`` by applying the :ref:`Multiple shooting ` construction with the updated window partition. In particular: @@ -185,7 +190,7 @@ provide a utility function for mapping parameters between stage problems. Partitioning measurements and time windows ------------------------------------------ -The above training approaches above require either splitting measurements into +The above training approaches require either splitting measurements into curriculum stages (curriculum learning) or partitioning the simulation time span into windows (multiple shooting and curriculum multiple shooting). We recommend that tools supporting these methods provide the splitting schemes @@ -203,5 +208,5 @@ point per stage). For multiple shooting, window intervals ``[t0_i, tf_i]`` must be defined. We recommend supporting automatic window construction (e.g., take ``nWindows`` as input and allocate windows based on unique measurement time points) as well as -user-specified intervals. As a basic sanity check, tools should ensure that +user-specified intervals. As a basic sense check, tools should ensure that each window contains at least one measurement. From 59e7cda2662a3378c243efa5e00d3dfcbf07fb9f Mon Sep 17 00:00:00 2001 From: sebapersson Date: Fri, 12 Jun 2026 09:36:33 +0100 Subject: [PATCH 5/6] Update description training strategies --- doc/format_overview.rst | 1 + doc/training_approaches.rst | 207 ---------------------------------- doc/training_strategies.rst | 216 ++++++++++++++++++++++++++++++++++++ 3 files changed, 217 insertions(+), 207 deletions(-) delete mode 100644 doc/training_approaches.rst create mode 100644 doc/training_strategies.rst diff --git a/doc/format_overview.rst b/doc/format_overview.rst index bc91522..20c2519 100644 --- a/doc/format_overview.rst +++ b/doc/format_overview.rst @@ -10,4 +10,5 @@ contribute to the development of PEtab SciML. Format Specification Supported Layers and Activation Functions + SciML Training strategies at the PEtab level Development process diff --git a/doc/training_approaches.rst b/doc/training_approaches.rst deleted file mode 100644 index eb6e673..0000000 --- a/doc/training_approaches.rst +++ /dev/null @@ -1,207 +0,0 @@ -SciML Training strategies at the PEtab level -============================================ - -Training (parameter estimating) SciML models can be challenging, and often -standard ML training workflows (e.g., training with Adam for a fixed number of -epochs) fail to find a good minimum or require many training epochs. - -Several training strategies have been developed to address this. These include -curriculum learning, multiple shooting, and combined curriculum multiple -shooting, all of which can be implemented at the PEtab abstraction level for -ODE models as well as hybrid PEtab SciML problems. This page describes these -PEtab-level abstractions for tool developers. The PEtab SciML library also -provides reference implementations. - -Curriculum learning -------------------- - -Curriculum learning is a training strategy where the training problem is -made progressively harder over successive curriculum stages. For PEtab -problems, a curriculum can be defined by gradually increasing the number of -measurement time points (and typically the simulation end time) over a fixed -number of stages. This can be implemented at the PEtab level as follows: - -Inputs: - -- A PEtab problem (PEtab v1 or v2). -- The number of curriculum stages, ``nStages``. -- A schedule ``n_i`` specifying how many measurements are included in stage - ``i``. - -1. Sort the measurement table in the input PEtab problem by the ``time`` - column. -2. Create ``nStages`` PEtab sub-problems by copying the input problem. For - stage ``i``, filter the time-sorted measurement table to keep the first - ``n_i`` measurements. -3. Optionally filter the condition, observable and experiment tables to only - include entries required by the measurement table for each sub-problem. - -A practical consideration for tools implementing and/or importing curriculum -problems is to keep parameter ordering consistent across stages, which -simplifies transferring parameters between stages. - -.. _multiple_shooting: - -Multiple shooting ------------------ - -In multiple shooting, the simulation time span of each PEtab experiment is -split into windows that are fitted jointly. Each window has its own estimated -initial state values, and a continuity penalty is introduced to encourage a -continuous trajectory between adjacent windows. This can be implemented at the -PEtab level as follows: - -Inputs: - -- A PEtab problem (PEtab v2). -- The number of multiple-shooting windows, ``nWindows``. -- A window partition ``[t0_i, tf_i]`` for each window ``i = 1..nWindows`` such - that the union of windows covers the full measurement time range, and - ``t0_i != tf_i`` for all windows. -- A continuity penalty parameter ``lambda``. - -1. Copy the input PEtab problem to create a multiple shooting (MS) PEtab - problem. -2. In the MS PEtab problem, add the penalty weight parameter ``lambda`` to the - parameter table as a non-estimated parameter and set an appropriate nominal - value. -3. For each PEtab experiment with ID ``expId`` in the MS PEtab problem: - 1. Create ``nWindows`` new PEtab experiments with IDs ``WINDOW{i}_{expId}`` - and set the initial time to ``t0_i`` for window ``i = 1..nWindows``. - 2. In the experiment table, remove the original experiment IDs and keep - only the windowed experiments. Assign each PEtab condition to the - corresponding window experiment. If a PEtab condition occurs at a - time point that lies in the overlap of windows ``i-1`` and ``i``, assign - the condition to experiment ``WINDOW{i-1}_{expId}``. - 3. In the measurement table, assign all measurements in the time interval - ``[t0_i, tf_i]`` for experiment ``expId`` to experiment - ``WINDOW{i}_{expId}``. If MS windows overlap at time points that contain - measurements, duplicate those measurements so they appear in each - relevant window. - 4. For each window ``i > 1`` such that there exists at least one - measurement for ``expId`` at time ``t >= t0_i`` in the original problem - (i.e., at least one subsequent window contains measurements), assign - initial window values and a continuity penalty: - 1. In the parameter table, create parameters - ``WINDOW{i}_{expId}_init_stateId{j}`` for each model state - ``stateId{j}``. Mark them as estimated and choose appropriate bounds. - 2. In the condition table, create a condition with ID - ``WINDOW{i}_{expId}_condition0`` that assigns each ``stateId{j}`` to - ``WINDOW{i}_{expId}_init_stateId{j}``. - 3. Assign condition ``WINDOW{i}_{expId}_condition0`` as the initial - condition for experiment ``WINDOW{i}_{expId}`` at time ``t0_i``. - 4. In the observable table, create an observable with ID - ``WINDOW{i}_{expId}_penalty_stateId{j}`` for each model state - ``stateId{j}`` and set - - - ``observableFormula = sqrt(lambda) * (stateId{j} - WINDOW{i}_{expId}_init_stateId{j})`` - - ``noiseFormula = 1.0`` - - ``noiseDistribution = normal`` - - 5. In the measurement table, add a row for experiment - ``WINDOW{i}_{expId}`` and observable - ``WINDOW{i}_{expId}_penalty_stateId{j}`` at time ``t0_i`` with - ``measurement = 0.0``. This yields an L2 (quadratic) penalty. - -Naive multiple shooting can perform poorly when states have different scales, -since a single penalty weight may be impossible to tune. In this case, a -log-scale penalty such as - -``sqrt(lambda) * (log(abs(stateId{j})) - log(WINDOW{i}_{expId}_init_stateId{j}))`` - -can be effective, where ``abs`` avoid potential problems with states going -below zero due to numerical errors. - -From a runtime performance perspective, the number of initial-window -parameters scales with the number of windows, states, and PEtab experiments, -which can be impractical for larger problems. Moreover, since initial-window -parameters must be estimated, this approach typically performs poorly for -partially observed systems; this is addressed by the curriculum multiple -shooting approach. - -Curriculum multiple shooting ----------------------------- - -Curriculum multiple shooting (CL+MS) combines multiple shooting with a -curriculum schedule. The idea is to start from a multiple-shooting formulation, -which is often easier to train, and then progressively reduce the number of -windows until the original (single-window) problem is recovered. This makes the -approach less sensitive to continuity-penalty tuning and ensures the final -parameters optimize the objective of the original PEtab problem. - -Practically, CL+MS defines ``nStages`` curriculum stages. Stage 1 corresponds -to a multiple-shooting problem with ``nWindows = nStages`` windows. In each -subsequent stage, the first ``nWindows-1`` windows are expanded to cover the -union of two adjacent windows, and the last window is dropped. This reduces -the number of windows by one per stage while increasing the time span covered -by each remaining window. The final stage has a single window and corresponds -to the original problem. This can be implemented at the PEtab level as follows: - -Inputs: - -- A PEtab problem (PEtab v2). -- The number of curriculum stages, ``nStages``. -- An initial window partition ``[t0_i, tf_i]`` for stage 1 with - ``i = 1..nStages``, such that the union of windows covers the full - measurement time range and ``t0_i != tf_i`` for all windows. -- A continuity penalty parameter ``lambda`` (used in the multiple-shooting - stages). - -1. Construct stage 1 as a multiple-shooting (MS) PEtab problem with - ``nWindows = nStages`` using the procedure in - :ref:`Multiple shooting `. -2. For curriculum stage ``k = 2..(nStages-1)``: - 1. Set the number of windows to ``nWindows = nStages - k + 1``. - 2. Define the MS window time spans for stage ``k`` by merging adjacent - windows from the previous stage: - - For ``i = 1..nWindows`` set ``t0_i^{(k)} = t0_i^{(k-1)}`` and - ``tf_i^{(k)} = tf_{i+1}^{(k-1)}``. - - Drop the last window of stage ``k-1``. - 3. Create the PEtab problem for stage ``k`` by applying the - :ref:`Multiple shooting ` construction with the - updated window partition. In particular: - - - Update the experiment table to contain only experiments - ``WINDOW{i}_{expId}`` for ``i = 1..nWindows``. - - Reassign and/or duplicate measurements to match - ``[t0_i^{(k)}, tf_i^{(k)}]``. - Measurements that in the original problem now appear in multiple - windows must be duplicated so they appear in each window. - - Include window-initial parameters and continuity-penalty observables - for windows ``i > 1`` as in multiple shooting. Note that the penalty - is applied at the initial time point of each window; in PEtab it is - not possible to define a continuity penalty over the full overlap - interval between two windows. - -3. The final stage corresponds to the original PEtab problem. Use the parameter - estimate from stage ``nStages-1`` to initialize optimization for the final - stage. - -A practical consideration for tools implementing and/or importing CL+MS is that -the number of window-initial parameters to estimate changes between stages. To -support transferring parameter values between stages, it can be beneficial to -provide a utility function for mapping parameters between stage problems. - -Partitioning measurements and time windows ------------------------------------------- - -The above training approaches above require either splitting measurements into -curriculum stages (curriculum learning) or partitioning the simulation time -span into windows (multiple shooting and curriculum multiple shooting). We -recommend that tools supporting these methods provide the splitting schemes -outlined below. - -For curriculum learning, the number of measurements per stage, ``n_i``, can be -chosen in two ways: (i) split by unique measurement time points and allocate -``n_i`` accordingly, or (ii) split by the total number of measurements, which -can be effective when there are few unique time points but many repeated -measurements. We recommend supporting both modes, as well as automatic -splitting (e.g., given ``nStages``, compute ``n_i`` for the user) and -user-defined schedules (e.g., explicit ``n_i`` per stage or a maximum time -point per stage). - -For multiple shooting, window intervals ``[t0_i, tf_i]`` must be defined. We -recommend supporting automatic window construction (e.g., take ``nWindows`` as -input and allocate windows based on unique measurement time points) as well as -user-specified intervals. As a basic sanity check, tools should ensure that -each window contains at least one measurement. diff --git a/doc/training_strategies.rst b/doc/training_strategies.rst new file mode 100644 index 0000000..8dcf4b9 --- /dev/null +++ b/doc/training_strategies.rst @@ -0,0 +1,216 @@ +SciML Training strategies at the PEtab level +============================================ + +Training (parameter estimating) SciML models can be challenging, and often +standard ML training workflows (e.g., training with Adam for a fixed number of +epochs) fail to find a good minimum or require many training epochs. + +Several training strategies have been developed to address this. These include +curriculum learning, multiple shooting, and combined curriculum multiple +shooting, all of which can be implemented at the PEtab abstraction level for +mechanistic ODE models as well as hybrid PEtab SciML problems. This page +describes these PEtab-level abstractions for tool developers. The PEtab SciML +library also provides reference implementations supporting both PEtab v2, and +PEtab SciML problems. + +Curriculum learning +------------------- + +Curriculum learning is a training strategy where the training problem is made +progressively harder over successive curriculum stages. For ODE-based models, a +curriculum can be defined by gradually increasing the measurement time horizon +over a fixed number of stages. This can be implemented at the PEtab level as +follows: + +Inputs: + +- A PEtab problem (PEtab v1 or v2). +- A schedule of stage end times ``t_end_i`` for ``i = 1..nStages`` such that + ``t_end_1 < t_end_2 < ... < t_end_nStages = max(measurement times)``. + +1. Create ``nStages`` PEtab sub-problems by copying the input problem. For + stage ``i``, filter the measurement table to keep only measurements with + time ``t <= t_end_i``. +2. Filter the experiment table to only include experiments referenced by the + filtered measurement table. Within each remaining experiment, drop + periods that start after ``t_end_i``. +3. Filter the condition table to only include conditions referenced by the + filtered experiment table. +4. Export each curriculum problem to disk, in directories named ``stage{i}`` + for ``i = 1..nStages``. + +A practical consideration for tools implementing and/or importing curriculum +problems is to ensure that parameters are transferred consistently across +stages in training loops. Although the number of estimated parameters does not +change between stages, different PEtab importers may use different internal +parameter orderings when importing the stage-problems, so a stage-aware mapping +is needed. + +.. _multiple_shooting: + +Multiple shooting +----------------- + +In multiple shooting, the simulation time span of each PEtab experiment is +split into windows that are fitted jointly. Each window has its own estimated +initial state values, and a continuity penalty is introduced to encourage a +continuous trajectory between adjacent windows. This can be implemented at the +PEtab level as follows: + +Inputs: + +- A PEtab problem (PEtab v2). If the problem has no experiment table, add a + default experiment to which all measurements are assigned. Problems with + pre-equilibration are not supported. +- A window partition ``[t0_i, tf_i]`` for each window such that the union of + windows covers the full measurement time range and ``t0_i != tf_i`` for all + windows. Adjacent windows may overlap. +- A continuity penalty parameter ``lambda``. +- An initial guess ``init_value`` for each estimated window initial state. + +1. Copy the input PEtab problem to create a multiple shooting (MS) PEtab + problem. For this problem, create empty measurement, experiment and + condition tables. +2. Add a non-estimated parameter ``MS_PENALTY_SQRT`` to the parameter table + with nominal value ``sqrt(lambda)``. +3. For each PEtab experiment with ID ``expId`` in the MS PEtab problem and + each window ``i = 1..nWindows``: + + 1. If the maximum measurement time of ``expId`` in the original problem + is strictly less than ``t0_i``, skip this window for this experiment + (no PEtab experiment is created and no measurements, parameters, + observables, or conditions are added). Otherwise, create a new PEtab + experiment with ID ``WINDOW{i}_EXPERIMENT_{expId}``. + 2. Build the PEtab conditions of ``WINDOW{i}_EXPERIMENT_{expId}``: + + - For window ``i = 1``, keep original conditions of ``expId`` that fall + in ``[t0_1, tf_1]``. If no condition is applied at ``t0_1``, add a + condition applied at ``t0_1`` with ``conditionId`` so the simulation + starts with the original PEtab problem initialization. + - For windows ``i > 1``, add a leading period at ``t0_i`` with the + window's IC condition (defined below). Keep original conditions that + are applied in ``[t0_i, tf_i]``. + + 3. Assign all measurements of ``expId`` in ``[t0_i, tf_i]`` to experiment + ``WINDOW{i}_EXPERIMENT_{expId}``. Measurements at exactly the boundary + between two adjacent windows are duplicated so they appear in both + windows. + 4. If ``i > 1``, add per-experiment window initial values and continuity + penalty: + + a. In the parameter table, create parameters + ``WINDOW{i}_EXPERIMENT_{expId}_PARAMETER_{stateId}`` for each model + state ``stateId``. Mark them as estimated, give them appropriate + bounds, and use ``init_value`` as the nominal value. + b. In the condition table, create a condition with ID + ``WINDOW{i}_EXPERIMENT_{expId}_IC`` that assigns each ``stateId`` to + ``WINDOW{i}_EXPERIMENT_{expId}_PARAMETER_{stateId}``. + c. In the observable table, create an observable with ID + ``WINDOW{i}_EXPERIMENT_{expId}_PENALTY_{stateId}`` for each model + state ``stateId`` and set + + - ``observableFormula = (stateId - WINDOW{i}_EXPERIMENT_{expId}_PARAMETER_{stateId}) * MS_PENALTY_SQRT`` + - ``noiseFormula = 1.0`` + - ``noiseDistribution = normal`` + + d. In the measurement table, add a row for experiment + ``WINDOW{i-1}_EXPERIMENT_{expId}`` and observable + ``WINDOW{i}_EXPERIMENT_{expId}_PENALTY_{stateId}`` at time + ``t0_i`` with ``measurement = 0.0``. This yields a quadratic (L2) + penalty evaluated where the simulated trajectory of window ``i-1`` + meets the estimated initial state of window ``i``. + +Note that all artifacts in step 3.4 are added per ``(window, experiment)`` +pair rather than globally per window, since trajectories differ between +experiments. ``MS_PENALTY_SQRT`` is added once and shared across all +experiments and windows. + +Naive multiple shooting can perform poorly when states have different scales, +since a single penalty weight may be impossible to tune. In this case, a +log-scale penalty such as + +``(log(abs(stateId)) - log(abs(WINDOW{i}_EXPERIMENT_{expId}_PARAMETER_{stateId}))) * MS_PENALTY_SQRT`` + +can be effective, where ``abs`` avoids potential problems with states going +below zero due to numerical errors. + +Curriculum multiple shooting +---------------------------- + +Curriculum multiple shooting (CMS) combines multiple shooting with a +curriculum schedule. The idea is to start from a multiple-shooting formulation, +which is often easier to train, and then progressively reduce the number of +windows until the original (single-window) problem is recovered. + +CMS defines ``nStages`` curriculum stages. Stage 1 is a multiple-shooting +problem with ``nStages`` windows. At each subsequent stage the last window is +dropped and every remaining window's end is shifted one position to the right; +equivalently, window ``i`` at stage ``k`` is ``[t0_i, tf_{i+k-1}]``. Each stage +therefore has one fewer window than the previous, with each remaining window +covering more of the time range. The final stage is a single window covering +``[t0_1, tf_nStages]`` and corresponds to the original problem. Stages 2 +onwards have overlapping windows; the multiple-shooting construction handles +this naturally when the continuity penalty is placed at ``t0_{i+1}``, the first +overlapping time point. The PEtab-level implementation is then: + +Inputs: + +- A PEtab problem (PEtab v2). +- An initial window partition ``[t0_i, tf_i]`` for stage 1 such that the union + of windows covers the full measurement time range and ``t0_i != tf_i`` for + all windows. The number of curriculum stages equals the number of windows + in this partition. +- A continuity penalty parameter ``lambda``. +- An initial guess ``init_value`` for each estimated window initial state. + +1. Construct stage 1 as a multiple-shooting (MS) PEtab problem with + ``nWindows = nStages`` using the procedure in + :ref:`Multiple shooting `. +2. For curriculum stage ``k = 2..(nStages-1)``: + + 1. Set the number of windows to ``nWindows = nStages - k + 1``. + 2. Define the stage-``k`` windows by dropping the last window from stage + ``k-1`` and extending the remaining windows. With the original window + starts ``t0_1, ..., t0_nStages`` and ends ``tf_1, ..., tf_nStages`` from + stage 1, the stage-``k`` windows are + + ``[t0_1, tf_k], [t0_2, tf_{k+1}], ..., [t0_{nStages-k+1}, tf_nStages]``. + + Note that windows now overlap pairwise. + 3. Create the PEtab problem for stage ``k`` by applying the + :ref:`Multiple shooting ` construction with the + updated window partition. Measurements falling in the overlap between + two windows are duplicated so they appear in each window. The continuity + penalty between windows ``i`` and ``i+1`` is placed at ``t0_{i+1}`` + (the first overlapping time point), evaluated in the experiment + ``WINDOW{i}_EXPERIMENT_{expId}``. + +3. The final stage (``k = nStages``) corresponds to the original PEtab problem. + Use the parameter estimate from stage ``nStages-1`` to initialize + optimization for the final stage. + +A practical consideration for tools implementing and/or importing CMS is that +the number of window-initial parameters to estimate changes between stages. To +support transferring parameter values between stages, it can be beneficial to +provide a utility function for mapping parameters between stage problems. + +Partitioning time windows +------------------------- + +The above training approaches above require either splitting measurements into +curriculum stages (curriculum learning) or partitioning the simulation time +span into windows (multiple shooting and curriculum multiple shooting). We +recommend that tools supporting these methods provide the splitting schemes +outlined below. + +For curriculum learning, splitting is done by unique measurement time points: +stage boundaries are placed at time points from the measurement table, and a +stage includes all measurements up to its boundary. We recommend supporting +both automatic splitting (e.g., given ``nStages``, compute stage boundaries for +the user) and user-defined schedules (e.g., explicit time points per stage). + +For multiple shooting, window intervals ``[t0_i, tf_i]`` must be defined. We +recommend supporting automatic window construction (e.g., take ``nWindows`` as +input and allocate windows based on unique measurement time points) as well as +user-specified intervals. As a basic sanity check, tools should ensure that +each window contains at least one measurement. 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Several training strategies have been developed to address this. These include curriculum learning, multiple shooting, and combined curriculum multiple @@ -19,32 +19,48 @@ Curriculum learning Curriculum learning is a training strategy where the training problem is made progressively harder over successive curriculum stages. For ODE-based models, a curriculum can be defined by gradually increasing the measurement time horizon -over a fixed number of stages. This can be implemented at the PEtab level as -follows: +over a fixed number of stages. + +.. container:: figure align-center + + .. image:: assets/cl_illustration.png + :alt: Hybrid model types + :class: only-light + + .. image:: assets/cl_illustration_dark.png + :alt: Hybrid model types + :class: only-dark + + .. container:: caption text-start + + **Curriculum learning illustration**. Each stage extends the measurement + time horizon, progressively adding more data points to train on. + +Curriculum learning can be implemented at the PEtab level as follows: Inputs: - A PEtab problem (PEtab v1 or v2). -- A schedule of stage end times ``t_end_i`` for ``i = 1..nStages`` such that - ``t_end_1 < t_end_2 < ... < t_end_nStages = max(measurement times)``. +- A schedule of stage end times :math:`t_{\text{end},i}` for + :math:`i = 1, \ldots, n_{\text{stages}}` such that + :math:`t_{\text{end},1} < t_{\text{end},2} < \cdots < t_{\text{end},n_{\text{stages}}} = \max(\text{measurement times})`. -1. Create ``nStages`` PEtab sub-problems by copying the input problem. For - stage ``i``, filter the measurement table to keep only measurements with - time ``t <= t_end_i``. +1. Create :math:`n_{\text{stages}}` PEtab sub-problems by copying the input + problem. For stage :math:`i`, filter the measurement table to keep only + measurements with time :math:`t \leq t_{\text{end},i}`. 2. Filter the experiment table to only include experiments referenced by the - filtered measurement table. Within each remaining experiment, drop - periods that start after ``t_end_i``. + filtered measurement table. Within each remaining experiment, drop periods + that start after :math:`t_{\text{end},i}`. 3. Filter the condition table to only include conditions referenced by the filtered experiment table. 4. Export each curriculum problem to disk, in directories named ``stage{i}`` - for ``i = 1..nStages``. + for :math:`i = 1, \ldots, n_{\text{stages}}`. A practical consideration for tools implementing and/or importing curriculum problems is to ensure that parameters are transferred consistently across stages in training loops. Although the number of estimated parameters does not change between stages, different PEtab importers may use different internal -parameter orderings when importing the stage-problems, so a stage-aware mapping -is needed. +parameter orderings when importing the stage-problems. .. _multiple_shooting: @@ -54,54 +70,75 @@ Multiple shooting In multiple shooting, the simulation time span of each PEtab experiment is split into windows that are fitted jointly. Each window has its own estimated initial state values, and a continuity penalty is introduced to encourage a -continuous trajectory between adjacent windows. This can be implemented at the -PEtab level as follows: +continuous trajectory between adjacent windows. + +.. container:: figure align-center + + .. image:: assets/ms_illustration.png + :alt: Hybrid model types + :class: only-light + :width: 75% + + .. image:: assets/ms_illustration_dark.png + :alt: Hybrid model types + :class: only-dark + :width: 75% + + .. container:: caption text-start + + **Multiple shooting illustration.** Each window is fitted with its own + initial state, and a continuity penalty encourages a continuous + trajectory across window boundaries. + + +Multiple shooting can be implemented at the PEtab level as follows: Inputs: - A PEtab problem (PEtab v2). If the problem has no experiment table, add a default experiment to which all measurements are assigned. Problems with pre-equilibration are not supported. -- A window partition ``[t0_i, tf_i]`` for each window such that the union of - windows covers the full measurement time range and ``t0_i != tf_i`` for all - windows. Adjacent windows may overlap. -- A continuity penalty parameter ``lambda``. -- An initial guess ``init_value`` for each estimated window initial state. +- A window partition :math:`[t_{0,i}, t_{f,i}]` for each window such that the + union of windows covers the full measurement time range and + :math:`t_{0,i} \neq t_{f,i}` for all windows. Adjacent windows may overlap. +- A continuity penalty parameter :math:`\lambda`. +- An initial guess :math:`v_0` for each estimated window initial state. 1. Copy the input PEtab problem to create a multiple shooting (MS) PEtab problem. For this problem, create empty measurement, experiment and condition tables. 2. Add a non-estimated parameter ``MS_PENALTY_SQRT`` to the parameter table - with nominal value ``sqrt(lambda)``. + with nominal value :math:`\sqrt{\lambda}`. 3. For each PEtab experiment with ID ``expId`` in the MS PEtab problem and - each window ``i = 1..nWindows``: + each window :math:`i = 1, \ldots, n_{\text{windows}}`: - 1. If the maximum measurement time of ``expId`` in the original problem - is strictly less than ``t0_i``, skip this window for this experiment + 1. If the maximum measurement time of ``expId`` in the original problem is + strictly less than :math:`t_{0,i}`, skip this window for this experiment (no PEtab experiment is created and no measurements, parameters, observables, or conditions are added). Otherwise, create a new PEtab experiment with ID ``WINDOW{i}_EXPERIMENT_{expId}``. 2. Build the PEtab conditions of ``WINDOW{i}_EXPERIMENT_{expId}``: - - For window ``i = 1``, keep original conditions of ``expId`` that fall - in ``[t0_1, tf_1]``. If no condition is applied at ``t0_1``, add a - condition applied at ``t0_1`` with ``conditionId`` so the simulation - starts with the original PEtab problem initialization. - - For windows ``i > 1``, add a leading period at ``t0_i`` with the - window's IC condition (defined below). Keep original conditions that - are applied in ``[t0_i, tf_i]``. - - 3. Assign all measurements of ``expId`` in ``[t0_i, tf_i]`` to experiment - ``WINDOW{i}_EXPERIMENT_{expId}``. Measurements at exactly the boundary - between two adjacent windows are duplicated so they appear in both - windows. - 4. If ``i > 1``, add per-experiment window initial values and continuity - penalty: + - For window :math:`i = 1`, keep original conditions of ``expId`` that + fall in :math:`[t_{0,1}, t_{f,1}]`. If no condition is applied at + :math:`t_{0,1}`, add a condition applied at :math:`t_{0,1}` with + ``conditionId`` so the simulation starts with the original PEtab + problem initialization. + - For windows :math:`i > 1`, add a leading period at :math:`t_{0,i}` + with the window's IC condition (defined below). Keep original + conditions that are applied in :math:`[t_{0,i}, t_{f,i}]`. + + 3. Assign all measurements of ``expId`` in :math:`[t_{0,i}, t_{f,i}]` to + experiment ``WINDOW{i}_EXPERIMENT_{expId}``. Measurements at exactly + the boundary between two adjacent windows are duplicated so they + appear in both windows. + 4. If :math:`i > 1`, add per-experiment window initial values and + continuity penalty: a. In the parameter table, create parameters ``WINDOW{i}_EXPERIMENT_{expId}_PARAMETER_{stateId}`` for each model state ``stateId``. Mark them as estimated, give them appropriate - bounds, and use ``init_value`` as the nominal value. + bounds, and use :math:`v_0` as the nominal value. b. In the condition table, create a condition with ID ``WINDOW{i}_EXPERIMENT_{expId}_IC`` that assigns each ``stateId`` to ``WINDOW{i}_EXPERIMENT_{expId}_PARAMETER_{stateId}``. @@ -116,14 +153,14 @@ Inputs: d. In the measurement table, add a row for experiment ``WINDOW{i-1}_EXPERIMENT_{expId}`` and observable ``WINDOW{i}_EXPERIMENT_{expId}_PENALTY_{stateId}`` at time - ``t0_i`` with ``measurement = 0.0``. This yields a quadratic (L2) - penalty evaluated where the simulated trajectory of window ``i-1`` - meets the estimated initial state of window ``i``. + :math:`t_{0,i}` with ``measurement = 0.0``. This yields a quadratic + (L2) penalty evaluated where the simulated trajectory of window + :math:`i-1` meets the estimated initial state of window :math:`i`. -Note that all artifacts in step 3.4 are added per ``(window, experiment)`` -pair rather than globally per window, since trajectories differ between -experiments. ``MS_PENALTY_SQRT`` is added once and shared across all -experiments and windows. +Note that all artifacts in step 3-4 are added per ``(window, experiment)``pair +rather than globally per window, since trajectories differ between experiments. +``MS_PENALTY_SQRT`` is added once and shared across all experiments and +windows. Naive multiple shooting can perform poorly when states have different scales, since a single penalty weight may be impossible to tune. In this case, a @@ -142,52 +179,73 @@ curriculum schedule. The idea is to start from a multiple-shooting formulation, which is often easier to train, and then progressively reduce the number of windows until the original (single-window) problem is recovered. -CMS defines ``nStages`` curriculum stages. Stage 1 is a multiple-shooting -problem with ``nStages`` windows. At each subsequent stage the last window is -dropped and every remaining window's end is shifted one position to the right; -equivalently, window ``i`` at stage ``k`` is ``[t0_i, tf_{i+k-1}]``. Each stage -therefore has one fewer window than the previous, with each remaining window -covering more of the time range. The final stage is a single window covering -``[t0_1, tf_nStages]`` and corresponds to the original problem. Stages 2 -onwards have overlapping windows; the multiple-shooting construction handles -this naturally when the continuity penalty is placed at ``t0_{i+1}``, the first -overlapping time point. The PEtab-level implementation is then: +.. container:: figure align-center + + .. image:: assets/cms_illustration.png + :alt: Hybrid model types + :class: only-light + + .. image:: assets/cms_illustration_dark.png + :alt: Hybrid model types + :class: only-dark + + .. container:: caption text-start + + **Curriculum multiple shooting illustration.** Each stage is a multiple + shooting problem with progressively fewer windows, until the final stage + recovers the original problem. + +CMS defines :math:`n_{\text{stages}}` curriculum stages. Stage 1 is a +multiple-shooting problem with :math:`n_{\text{stages}}` windows. At each +subsequent stage the last window is dropped and every remaining window's end +is shifted one position to the right; equivalently, window :math:`i` at stage +:math:`k` is :math:`[t_{0,i}, t_{f,i+k-1}]`. Each stage therefore has one fewer +window than the previous, with each remaining window covering more of the time +range. The final stage is a single window covering +:math:`[t_{0,1}, t_{f,n_{\text{stages}}}]` and corresponds to the original +problem. Stages 2 onwards have overlapping windows; the multiple-shooting +construction handles this naturally when the continuity penalty is placed at +:math:`t_{0,i+1}`, the first overlapping time point. The PEtab-level +implementation is then: Inputs: - A PEtab problem (PEtab v2). -- An initial window partition ``[t0_i, tf_i]`` for stage 1 such that the union - of windows covers the full measurement time range and ``t0_i != tf_i`` for - all windows. The number of curriculum stages equals the number of windows - in this partition. -- A continuity penalty parameter ``lambda``. -- An initial guess ``init_value`` for each estimated window initial state. +- An initial window partition :math:`[t_{0,i}, t_{f,i}]` for stage 1 such that + the union of windows covers the full measurement time range and + :math:`t_{0,i} \neq t_{f,i}` for all windows. The number of curriculum + stages equals the number of windows in this partition. +- A continuity penalty parameter :math:`\lambda`. +- An initial guess :math:`v_0` for each estimated window initial state. 1. Construct stage 1 as a multiple-shooting (MS) PEtab problem with - ``nWindows = nStages`` using the procedure in + :math:`n_{\text{windows}} = n_{\text{stages}}` using the procedure in :ref:`Multiple shooting `. -2. For curriculum stage ``k = 2..(nStages-1)``: +2. For curriculum stage :math:`k = 2, \ldots, n_{\text{stages}} - 1`: - 1. Set the number of windows to ``nWindows = nStages - k + 1``. - 2. Define the stage-``k`` windows by dropping the last window from stage - ``k-1`` and extending the remaining windows. With the original window - starts ``t0_1, ..., t0_nStages`` and ends ``tf_1, ..., tf_nStages`` from - stage 1, the stage-``k`` windows are + 1. Set the number of windows to + :math:`n_{\text{windows}} = n_{\text{stages}} - k + 1`. + 2. Define the stage-:math:`k` windows by dropping the last window from + stage :math:`k - 1` and extending the remaining windows. With the + original window starts :math:`t_{0,1}, \ldots, t_{0,n_{\text{stages}}}` + and ends :math:`t_{f,1}, \ldots, t_{f,n_{\text{stages}}}` from stage 1, + the stage-:math:`k` windows are - ``[t0_1, tf_k], [t0_2, tf_{k+1}], ..., [t0_{nStages-k+1}, tf_nStages]``. + :math:`[t_{0,1}, t_{f,k}], [t_{0,2}, t_{f,k+1}], \ldots, [t_{0,n_{\text{stages}}-k+1}, t_{f,n_{\text{stages}}}]`. Note that windows now overlap pairwise. - 3. Create the PEtab problem for stage ``k`` by applying the + 3. Create the PEtab problem for stage :math:`k` by applying the :ref:`Multiple shooting ` construction with the updated window partition. Measurements falling in the overlap between two windows are duplicated so they appear in each window. The continuity - penalty between windows ``i`` and ``i+1`` is placed at ``t0_{i+1}`` - (the first overlapping time point), evaluated in the experiment - ``WINDOW{i}_EXPERIMENT_{expId}``. + penalty between windows :math:`i` and :math:`i+1` is placed at + :math:`t_{0,i+1}` (the first overlapping time point), evaluated in the + experiment ``WINDOW{i}_EXPERIMENT_{expId}``. -3. The final stage (``k = nStages``) corresponds to the original PEtab problem. - Use the parameter estimate from stage ``nStages-1`` to initialize - optimization for the final stage. +3. The final stage (:math:`k = n_{\text{stages}}`) corresponds to the original + PEtab problem. Use the parameter estimate from stage + :math:`n_{\text{stages}} - 1` to initialize optimization for the final + stage. A practical consideration for tools implementing and/or importing CMS is that the number of window-initial parameters to estimate changes between stages. To @@ -206,11 +264,13 @@ outlined below. For curriculum learning, splitting is done by unique measurement time points: stage boundaries are placed at time points from the measurement table, and a stage includes all measurements up to its boundary. We recommend supporting -both automatic splitting (e.g., given ``nStages``, compute stage boundaries for -the user) and user-defined schedules (e.g., explicit time points per stage). - -For multiple shooting, window intervals ``[t0_i, tf_i]`` must be defined. We -recommend supporting automatic window construction (e.g., take ``nWindows`` as -input and allocate windows based on unique measurement time points) as well as -user-specified intervals. As a basic sanity check, tools should ensure that -each window contains at least one measurement. +both automatic splitting (e.g., given :math:`n_{\text{stages}}`, compute stage +boundaries for the user) and user-defined schedules (e.g., explicit time +points per stage). + +For multiple shooting, window intervals :math:`[t_{0,i}, t_{f,i}]` must be +defined. We recommend supporting automatic window construction (e.g., take +:math:`n_{\text{windows}}` as input and allocate windows based on unique +measurement time points) as well as user-specified intervals. As a basic +sanity check, tools should ensure that each window contains at least one +measurement.