FNO Darcy Flow — AutoResearch Benchmark

A self-contained Fourier Neural Operator (FNO) benchmark for 2-D Darcy Flow, adapted from raj-brown/fourier_neural_operator (Li et al., 2020 — "Fourier Neural Operator for Parametric Partial Differential Equations").

Used as the target task in the sciml_auto_research automated research framework.

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Task

Learn the solution operator of the 2-D Darcy Flow equation:

−∇ · (a(x)∇u(x)) = f(x),   x ∈ (0,1)²
u(x) = 0,                    x ∈ ∂(0,1)²

• Input a(x): permeability field sampled from a log-normal push-forward of a Gaussian random field
  (ψ#N(0,(−Δ+9I)⁻²), values 12 on positive part, 3 on negative)
• Output u(x): PDE solution via 2nd-order finite differences on a 421×421 grid
• Evaluation resolution: 85×85 (downsampled by factor 5 from 421)

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Paper Baseline (Table 4, FNO paper)

Parameter	Value
Fourier modes	12 × 12
Hidden width	32
Layers	4
Optimizer	Adam
Learning rate	1e-3, halved every 100 epochs
Epochs	500
Train samples	1000
Test samples	100
Reported test L2 rel. error	~0.0108

For AutoResearch runs, epochs=50 is used for faster iteration.

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Dataset

Official FNO paper dataset (required):

File	Size	Content
piececonst_r421_N1024_smooth1.mat	~1.6 GB	1024 training samples, 421×421
piececonst_r421_N1024_smooth2.mat	~1.6 GB	1024 test samples, 421×421

Download

pip install gdown

# Training data
gdown --id 1ViDqN7nc_VCnMackiXv_d7CHZANAFKzV -O piececonst_r421_N1024_smooth1.mat

# Test data
gdown --id 1ViDqN7nc_VCnMackiXv_d7CHZANAFKzV -O piececonst_r421_N1024_smooth2.mat

Alternatively, download Darcy_421.zip from the FNO Google Drive and extract both .mat files.

Set data_root in configs/train_config.yaml to the directory containing both .mat files.

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File Structure

fno_demo/
├── prepare.py               # Frozen evaluation harness (do not modify)
├── train.py                 # Mutable training script (AutoResearch modifies this)
├── configs/
│   ├── train_config.yaml    # Tunable hyperparameters
│   └── autoresearch_421.yaml
└── program.md               # AutoResearch agent instructions

prepare.py (frozen)

• Loads .mat files via MatReader
• Applies UnitGaussianNormalizer to both a(x) (input) and u(x) (output)
• Appends 2-D grid coordinates to input (3 channels total)
• Exposes get_dataloaders() and evaluate() with y_normalizer for decoding predictions back to physical space

train.py (mutable)

• SpectralConv2d + FNO2d defined inline (agents may modify architecture)
• Uses torch.fft.rfft2 / irfft2 (PyTorch ≥ 1.8)
• Decodes both prediction and target via y_normalizer.decode() before computing loss
• Prints FINAL_METRICS_START / FINAL_METRICS_END block for framework parsing

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Quick Start

conda activate cfd-gpu

# Edit data_root in configs/train_config.yaml first
python train.py --config configs/train_config.yaml

AutoResearch

cd ../   # sciml_auto_research root
bash run_claude.sh   # Claude (Anthropic)
bash run_codex.sh    # Codex / GPT-4.1 (OpenAI)

Results are saved to results/, logs to logs/, plots to images/.