Physics-Informed Networks and Neural Operators for PDEs

Two ways of learning solutions to PDEs with neural networks, implemented from scratch in PyTorch:

1. Physics-informed neural networks (PINNs) that solve a 2D linear-elasticity problem, a plate with a circular hole under tension, using only the governing equations and boundary conditions. A second variant adds 50 sparse displacement measurements.
2. Neural operators that learn the map from permeability field to pressure field for 2D Darcy flow, comparing a plain CNN with a Fourier Neural Operator (FNO).

_{Fourier Neural Operator on a held-out Darcy-flow sample: permeability a(x), true and predicted pressure u(x), and absolute error (below 5 × 10⁻⁴ against a peak pressure of about 0.29).}

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Results

Model	Task	Metric	Train	Test
CNN	Darcy flow, 32×32	relative L² error	4.9 × 10⁻²	6.1 × 10⁻²
FNO	Darcy flow, 32×32	relative L² error	3.0 × 10⁻³	2.1 × 10⁻³
PINN (physics only)	Plate with hole	max |ux − uxFEM|	—	≈ 4.2 × 10⁻²
PINN + 50 measurements	Plate with hole	max |ux − uxFEM|	—	≈ 5.6 × 10⁻³

All four runs are the final configurations: 200 epochs for the operators, 50 000 Adam iterations for the PINNs. For scale, the FEM displacement u_x peaks at about 0.06.

Takeaways

• On the same data, the FNO has ~30× lower test error than the CNN. Mixing globally in Fourier space suits the elliptic Darcy operator, where each output value depends on the whole input field.
• The physics-only PINN reaches a low residual loss (~10⁻⁵) but converges to a displacement field that is visibly wrong. It also underestimates the stress concentration at the hole. Adding just 50 measured displacement points anchors the solution and cuts the peak error by ~8×.

Plate with a hole: PINN vs FEM

Physics-only PINN:

Data-assisted PINN (50 measurements):

Darcy flow: CNN baseline

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Method

PINN for plane-stress elasticity

The domain is a quarter plate with a circular hole, using symmetry boundary conditions. A traction of σ·n = (0.1, 0) is applied on the right edge; the top edge and the hole are traction-free. The material has E = 10 and ν = 0.3.

Two MLPs (3 × 128 hidden units, tanh, float64) are trained jointly:

• disp_net: (x, y) → (u_x, u_y)
• stress_net: (x, y) → (σ₁₁, σ₂₂, σ₁₂)

The loss is a sum of MSE terms, with all derivatives computed by autograd:

• Constitutive consistency: stress_net must match Hooke's law applied to the strains of disp_net, at interior and boundary points.
• Equilibrium: ∇·σ = 0.
• Boundary conditions: symmetry displacements, applied traction, and a traction-free hole.
• Measurements (pinn_data.py only): MSE against 50 randomly chosen FEM displacement values, weighted ×100.

Training uses Adam (learning rate 10⁻³) with StepLR (×0.5 every 2000 iterations). Ground truth comes from a MATLAB FE solver meshed with DistMesh.

Operator learning for Darcy flow

The goal is to learn the solution operator 𝒢: a(x) ↦ u(x) of −∇·(a∇u) = f on the unit square. The dataset has 1000 training and 100 test pairs on a 32×32 grid.

• CNN: six same-padding convolutions (64 channels, GELU) that map the input to the output at the same resolution.
• FNO: lifts (a, x, y) to 32 channels, then applies 4 Fourier layers. Each layer is a spectral convolution keeping the lowest 12×12 modes plus a pointwise channel MLP and a 1×1 skip path. A final MLP projects back to one channel.

Both models use unit-Gaussian input normalisation, a relative L² loss, Adam (learning rate 10⁻³) with StepLR (×0.7 every 50 epochs), and batch size 20.

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Repository structure

.
├── pinn.py                    # Physics-only PINN (plate elasticity)
├── pinn_data.py               # PINN + sparse measurements (plate elasticity)
├── darcy_cnn.py               # CNN operator baseline (Darcy flow)
├── darcy_fno.py               # Fourier Neural Operator (Darcy flow)
├── sciml/
│   ├── pinn.py                # Networks, physics losses, training loop, plotting
│   └── darcy.py               # CNN, FNO, data loading, training loop, plotting
├── plate_fem/                 # MATLAB FE solver + DistMesh; generates plate_data.mat
├── darcy_data/                # Place the Darcy .mat datasets here (not tracked)
└── assets/                    # Figures used in this README

Reproducing

Dependencies are managed with uv:

uv sync

Data

• Plate problem: plate_fem/plate_data.mat is included. To regenerate it, run plate_fem/Plate_hole.m in MATLAB.
• Darcy flow: the datasets are not redistributed here. Place Darcy_2D_data_train.mat and Darcy_2D_data_test.mat in darcy_data/. They are HDF5 .mat files with 32×32×N a_field and u_field arrays.

Train

uv run pinn.py --iterations 50000
uv run pinn_data.py --iterations 50000 --measurement-weight 100
uv run darcy_cnn.py --epochs 200
uv run darcy_fno.py --epochs 200 --modes 12 --width 32

Each script writes its loss curves, prediction plots, a CSV history and checkpoints to outputs/<run>/. Pass --resume-from outputs/<run>/checkpoint_latest.pt to continue a run, and --help to list all options. Training uses CUDA when it is available.

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Acknowledgements

• Originally developed for 4C11 Data-Driven and Learning-Based Methods in Mechanics and Materials, Department of Engineering, University of Cambridge, which provided the problem setup, FE solver scaffolding and datasets.
• DistMesh © Per-Olof Persson, GPL v2+. It is vendored unmodified in plate_fem/distmesh/ under its own licence.
• FNO architecture after Li et al., Fourier Neural Operator for Parametric Partial Differential Equations, ICLR 2021.

Licence

The code written for this project is licensed under the MIT License. Third-party components keep their original licences.