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Newton Informed Neural Operator for Solving Nonlinear Partial Differential Equations
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Proposes the Newton Informed Neural Operator (NINO), a DeepONet-based surrogate that learns one Newton step for nonlinear elliptic/reaction-diffusion PDEs, trained with a mix of supervised MSE and a physics-based Newton residual loss to find multiple solutions with less labeled data.
Problem
Solving nonlinear PDEs with multiple solutions is hard for PINNs and neural operators (ill-posed, one solution per training run), and classical Newton solvers are expensive for large-scale or batched initial conditions.
Approach
Learn the Newton update operator G(u)=δu with a DeepONet trained by combining a supervised MSE loss on a small labeled set with an unsupervised Newton-residual loss (L − f'(u))O(u) + (Lu − f(u)) on many unlabeled initial states, so that iterating the learned operator converges to multiple solutions and is fast at inference.
Key results
- On the convex case, DeepONet trained with Newton loss (method 2) shows substantially lower test L2 and H1 error than MSE-only training on 500 supervised samples.
- Batched NINO inference takes 1.1E-4 s for 500 Newton systems and 1.4E-4 s for 5000, versus 31.52 s and 321.15 s for the GPU Newton solver (float32).
- Newton-loss-trained operator maps an unseen ring-like initial state to a steady state, showing it can find solutions absent from supervised data.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace the classical Newton solver with NINO | solving 500 linear Newton systems · Non-convex PDE, 500 linear Newton systems | execution time (s): 31.52 -> 1.1E-4 | Helped |
| replace the classical Newton solver with NINO | solving 5000 linear Newton systems · Non-convex PDE, 5000 linear Newton systems | execution time (s): 321.15 -> 1.4E-4 | Helped |
| add Newton loss to the MSE-only training objective (ablation) | DeepONet (method 1) · Convex PDE | test L2 error | Helped |
| add Newton loss to the MSE-only training objective (ablation) | DeepONet (method 1) · Convex PDE | test H1 error | Helped |
| add Newton loss to the MSE-only training objective (ablation) | DeepONet (method 1) · Non-convex PDE with multiple solutions | test error | Helped |
| add Newton loss and unsupervised data to training (ablation) | DeepONet (method 1) · Gray-Scott model | absolute L2 test error | Helped |
Surprising findings
- Inference time for NINO barely grows from 500 to 5000 systems, while the classical solver's time scales roughly tenfold.
Limitations
- Theory restricts approximation to local regions where the linearized problem is uniquely solvable (Assumption 1(i)); global operator learning is ill-posed with multiple solutions.
- Experiments are limited to 2D problems on small grids (63x63) with a DeepONet variant and fixed hyperparameters.
- Requires a classical solver to generate supervised solutions and Jacobian-based Newton loss, so unsupervised training still needs PDE coefficients and derivatives.
Experimental setup
Datasets: synthetic Newton iteration sequences from 2D convex and non-convex PDEs Gray-Scott steady-state data generated by finite difference and Newton solver
Benchmarks: 2D convex Poisson-type problem 2D non-convex problem with multiple solutions Gray-Scott model
Metrics: L2 test error H1 test error execution time average convergence rate
Baselines: DeepONet-MSE classical Newton solver (GPU, CuPy) PINN HomPINNs
Builds on: DeepONet PINN Newton method HomPINNs
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