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Learning a Neural Solver for Parametric PDEs to Enhance Physics-Informed Methods
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Proposes a learned neural solver that transforms physics-informed loss gradients (conditioned on PDE parameters) with a few iterations, enabling fast test-time solving of parametric PDEs.
Problem
Physics-informed deep learning optimization is ill-conditioned and slow, requiring many iterations and retraining per PDE instance; vanilla PINNs solve only one instance at a time.
Approach
Learn a neural network F that transforms the SGD gradient of the PDE residual loss, conditioned on PDE parameters (coefficients, forcing, initial/boundary conditions), so that a few (2-5) iterations over a B-spline ansatz reach accurate solutions for unseen parametric PDE instances; trained with a data loss against ground-truth solutions and shown to act as a learned preconditioner.
Key results
- On the test set, the neural solver achieves relative MSE 2.41e-2 on Helmholtz and 5.56e-5 on Poisson, beating all unsupervised and hybrid baselines by about an order of magnitude in most cases.
- At test time, the learned solver converges in about 5 steps on Poisson, while SGD, Adam, and L-BFGS fail to converge within 10,000 steps.
- Theory under linear assumptions claims the learned conditioner gives kappa(PA)=1, reducing steps from O(kappa(A) ln 1/eps) to O(ln 1/eps).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use the neural solver instead of MLP+basis | Parametric PDE solving · Helmholtz test | Relative MSE: 4.66e-2 -> 2.41e-2 | Helped |
| use the neural solver instead of PINO | Parametric PDE solving · Poisson test | Relative MSE: 2.80e-3 -> 5.56e-5 | Helped |
| use the neural solver instead of MLP+basis | Parametric PDE solving · NLRD test | Relative MSE: 2.85e-4 -> 2.91e-4 | Hurt |
| use the neural solver instead of MLP+basis | Parametric PDE solving · Darcy-Flow test | Relative MSE: 3.56e-2 -> 1.87e-2 | Helped |
| use the neural solver instead of PINO | Parametric PDE solving · Heat test | Relative MSE: 9.09e-3 -> 2.31e-3 | Helped |
| use the neural solver instead of classical optimizers (SGD, Adam, L-BFGS) | Test-time optimization on new Poisson equations · Poisson | optimization steps to converge: null | Helped |
Surprising findings
- The supervised MLP+basis baseline is strongest on NLRD (2.85e-4), beating the proposed method there.
- Hybrid PINO achieves lower error than the proposed method on Heat's comparisons among some settings, showing the method is not uniformly best.
Limitations
- Training requires more memory due to backpropagation through iterations, challenging in higher-dimensional bases.
- Theory assumes linearity and does not extend rigorously to nonlinear PDEs.
- Restricted to linear combinations of predefined bases (B-splines); neural-network ansatz increases ill-conditioning.
Experimental setup
Datasets: Helmholtz (generated) Poisson (generated) Reaction-Diffusion (NLRD) Darcy Flow Heat Advection (PDEBench)
Benchmarks: Helmholtz Poisson NLRD Darcy Flow Heat
Metrics: relative MSE convergence steps
Baselines: MLP+basis PINNs+L-BFGS PINNs-multi-opt PPINNs P2INNs PO-DeepONet PI-DeepONet PINO
Builds on: PINNs FNO DeepONet PINO L-BFGS
Closest papers by meaning
- An operator preconditioning perspective on training in physics-informed machine learning (ICLR 2024 · similarity 0.90)
- Parameterized Physics-informed Neural Networks for Parameterized PDEs (ICML 2024 · similarity 0.90)
- Text2PDE: Latent Diffusion Models for Accessible Physics Simulation (ICLR 2025 · similarity 0.89)
- Boosting Generalization in Parametric PDE Neural Solvers through Adaptive Conditioning (NeurIPS 2024 · similarity 0.89)
- Number Theoretic Accelerated Learning of Physics-Informed Neural Networks (AAAI 2025 · similarity 0.89)
- PRDP: Progressively Refined Differentiable Physics (ICLR 2025 · similarity 0.89)
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