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Mechanistic PDE Networks for Discovery of Governing Equations
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Proposes Mechanistic PDE Networks that embed a GPU-parallel, sparse, differentiable multigrid solver for linear PDEs inside a neural network to discover governing PDEs from noisy spatiotemporal data.
Problem
Discovering governing partial differential equations from spatiotemporal data that are nonlinear, noise-robust, and expressible with flexible differentiable terms, while solving the resulting PDEs in a memory-efficient way.
Approach
Represent spatiotemporal data as space-time varying linear PDEs inside a neural network, solve them with a differentiable multigrid-preconditioned FGMRES solver (NeuRLP-PDE) via a KKT saddle-point formulation, and parameterize PDE expression inputs with neural networks so that nonlinear PDEs can be discovered with sparsity-regularized coefficients.
Key results
- On the harder Ginzburg-Landau reaction-diffusion dataset, PDEFIND and WeakSINDy fail to find significant nonzero terms even on clean data, while MechNN-PDE keeps high TPR up to 40% noise, dropping to 0.6 at 80% noise.
- On the easy spiral reaction-diffusion dataset, all methods recover nonzero terms with TPR of 1 on clean data; PDEFIND degrades most with noise (TPR 0-0.2 at high noise).
- For the porous medium equation, the method recovers the exponent as 2.64 ± 0.03 versus the true value 2.675 on clean data.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use MechNN-PDE instead of PDEFIND on the easy reaction-diffusion dataset as noise increases | Reaction-diffusion discovery · Easy reaction-diffusion | TPR: TPR drops to 0-0.2 for PDEFIND; MechNN-PDE and WeakSINDy have similar TPR | Mixed |
| Use MechNN-PDE instead of PDEFIND or WeakSINDy | Reaction-diffusion discovery · Harder reaction-diffusion | TPR: TPR dropping to 0.6 at 80\% noise | Helped |
| Use MechNN-PDE instead of WeakSINDy | Navier-Stokes discovery · Incompressible Navier-Stokes | TPR | Helped |
| Increase Laplace grid resolution | NeuRLP-PDE · 2D Laplace | L2 error: 10^{-1}, 5\times 10^{-2} and 10^{-2} | Helped |
| Add the parameterized neural network basis input (ablation) | MechNN-PDE · PDE discovery | discovery flexibility | Helped |
| Use a two-layer MLP with a learnable input instead of a flat parameter vector (ablation) | MechNN-PDE · Reaction-diffusion discovery | training | Helped |
| Use an L1 final loss (ablation) | MechNN-PDE · Reaction-diffusion discovery | discovery | Helped |
| Repeat optimization with thresholded parameters (ablation) | MechNN-PDE · PDE discovery | coefficient accuracy | Helped |
Surprising findings
- PDEFIND and WeakSINDy fail to recover the harder reaction-diffusion equations even on noise-free data.
- WeakSINDy fails to discover the Navier-Stokes system while MechNN-PDE recovers the viscosity and nonzero terms, though with false positives.
Limitations
- Grid-based PDE methods have scalability limits in high dimensions.
- The solver sometimes sacrifices accuracy for speed.
- Identifiability of discovered PDEs is not analyzed.
Experimental setup
Datasets: Ginzburg-Landau reaction-diffusion spiral data (spectral method) Ginzburg-Landau reaction-diffusion data generated with Basilisk Incompressible Navier-Stokes data (viscosity 0.001) 1D diffusion, viscous and inviscid Burger's, porous medium data
Benchmarks: Ginzburg-Landau reaction-diffusion (easy and harder) Navier-Stokes 1D diffusion and Burgers Laplace equation solver unit test
Metrics: true positivity ratio (TPR) maximum relative coefficient error E_infinity L2 error relative error time and GPU memory
Baselines: PDEFIND WeakSINDy
Builds on: Mechanistic Neural Networks PDEFIND WeakSINDy PDE-LEARN
Closest papers by meaning
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