Enforcing governing equation constraints in neural PDE solvers via training-free projections

• URL: http://arxiv.org/abs/2511.17258v1
• Date: Fri, 21 Nov 2025 14:03:28 GMT
• Title: Enforcing governing equation constraints in neural PDE solvers via training-free projections
• Abstract summary: We evaluate two training-free, post hoc projections of approximate solutions using Jacobian-vector and vector-Jacobian products.<n>We find that both projections substantially reduce violations and improve accuracy over physics-informed baselines.
• Score: 5.539946449743145
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Neural PDE solvers used for scientific simulation often violate governing equation constraints. While linear constraints can be projected cheaply, many constraints are nonlinear, complicating projection onto the feasible set. Dynamical PDEs are especially difficult because constraints induce long-range dependencies in time. In this work, we evaluate two training-free, post hoc projections of approximate solutions: a nonlinear optimization-based projection, and a local linearization-based projection using Jacobian-vector and vector-Jacobian products. We analyze constraints across representative PDEs and find that both projections substantially reduce violations and improve accuracy over physics-informed baselines.

Related papers

• A Systematic Analysis of Automatic Differentiation versus Discretization-based Constraints for Physics-Informed PDE Solvers [5.821193246951147]
  Physics-informed neural networks (PINNs) represent a growing frontier using artificial intelligence to solve partial differential equations (PDEs)<n>We conduct systematic experiments across a wide spectrum of problems, from simple linear Poisson to high-Mach hypersonic flows with strong discontinuities.<n>Our results reveal a consistent trend: as nonlinearity strengthens, the accuracy advantage of discretization-based constraints becomes increasingly pronounced.
  arXiv  Detail & Related papers  (2026-09-07T12:44:35Z)
• SNAP-FM: Sparse Nonlinear Accelerated Projection for Physics-Constrained Generative Modeling [3.2297343473068882]
  Constrained sampling closes this gap, enforcing constraints exactly at inference time without retraining.<n>Standard ML frameworks obscure the structure that physical constraints induce naturally.<n>We exploit this bottleneck by exploiting the structure that sample-wise and local PDE couplings induce in the projection subproblems.
  arXiv  Detail & Related papers  (2026-06-30T19:35:09Z)
• BEACONS: Bounded-Error, Algebraically-Composable Neural Solvers for Partial Differential Equations [0.0]
  We show how it is possible to circumvent limitations by constructing formally-verified neural network solvers for PDEs.<n>We show how it is possible to construct rigorous extrapolatory bounds on the worst-case Linf errors of shallow neural network approximations.<n>The resulting framework, called BEACONS, comprises both an automatic code-proving for the neural solvers themselves, as well as a bespoke automated theorem-generator system for producing machine-checkable certificates of correctness.
  arXiv  Detail & Related papers  (2026-02-16T15:49:19Z)
• High precision PINNs in unbounded domains: application to singularity formulation in PDEs [83.50980325611066]
  We study the choices of neural network ansatz, sampling strategy, and optimization algorithm.<n>For 1D Burgers equation, our framework can lead to a solution with very high precision.<n>For the 2D Boussinesq equation, we obtain a solution whose loss is $4$ digits smaller than that obtained in citewang2023asymptotic with fewer training steps.
  arXiv  Detail & Related papers  (2025-06-24T02:01:44Z)
• Unisolver: PDE-Conditional Transformers Towards Universal Neural PDE Solvers [53.79279286773326]
  We present Unisolver, a novel Transformer model trained on diverse data and conditioned on diverse PDEs.<n>Unisolver achieves consistent state-of-the-art on three challenging large-scale benchmarks, showing impressive performance and generalizability.
  arXiv  Detail & Related papers  (2024-05-27T15:34:35Z)
• A physics-informed neural network framework for modeling obstacle-related equations [3.687313790402688]
  Physics-informed neural networks (PINNs) are an attractive tool for solving partial differential equations based on sparse and noisy data. Here we extend PINNs to solve obstacle-related PDEs which present a great computational challenge. The performance of the proposed PINNs is demonstrated in multiple scenarios for linear and nonlinear PDEs subject to regular and irregular obstacles.
  arXiv  Detail & Related papers  (2023-04-07T09:22:28Z)
• Solving High-Dimensional PDEs with Latent Spectral Models [74.1011309005488]
  We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space. LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks.
  arXiv  Detail & Related papers  (2023-01-30T04:58:40Z)
• Investigating and Mitigating Failure Modes in Physics-informed Neural Networks (PINNs) [0.0]
  This paper explores the difficulties in solving partial differential equations (PDEs) using physics-informed neural networks (PINNs) PINNs use physics as a regularization term in objective function. However, this approach is impractical in the absence of data or prior knowledge of the solution. Our findings demonstrate that high-order PDEs contaminate backpropagated gradients and hinder convergence.
  arXiv  Detail & Related papers  (2022-09-20T20:46:07Z)
• Bi-level Physics-Informed Neural Networks for PDE Constrained Optimization using Broyden's Hypergradients [29.487375792661005]
  We present a novel bi-level optimization framework to solve PDE constrained optimization problems. For the inner loop optimization, we adopt PINNs to solve the PDE constraints only. For the outer loop, we design a novel method by using Broyden'simat method based on the Implicit Function Theorem.
  arXiv  Detail & Related papers  (2022-09-15T06:21:24Z)
• Learning differentiable solvers for systems with hard constraints [48.54197776363251]
  We introduce a practical method to enforce partial differential equation (PDE) constraints for functions defined by neural networks (NNs) We develop a differentiable PDE-constrained layer that can be incorporated into any NN architecture. Our results show that incorporating hard constraints directly into the NN architecture achieves much lower test error when compared to training on an unconstrained objective.
  arXiv  Detail & Related papers  (2022-07-18T15:11:43Z)
• Lie Point Symmetry Data Augmentation for Neural PDE Solvers [69.72427135610106]
  We present a method, which can partially alleviate this problem, by improving neural PDE solver sample complexity. In the context of PDEs, it turns out that we are able to quantitatively derive an exhaustive list of data transformations. We show how it can easily be deployed to improve neural PDE solver sample complexity by an order of magnitude.
  arXiv  Detail & Related papers  (2022-02-15T18:43:17Z)
• Physics-Informed Neural Operator for Learning Partial Differential Equations [55.406540167010014]
  PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families.
  arXiv  Detail & Related papers  (2021-11-06T03:41:34Z)
• POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition [0.0]
  Deep learning-based reduced order models (DL-ROMs) have been recently proposed to overcome common limitations shared by conventional reduced order models (ROMs) In this paper we propose a possible way to avoid an expensive training stage of DL-ROMs, by (i) performing a prior dimensionality reduction through POD, and (ii) relying on a multi-fidelity pretraining stage. The proposed POD-DL-ROM is tested on several (both scalar and vector, linear and nonlinear) time-dependent parametrized PDEs.
  arXiv  Detail & Related papers  (2021-01-28T07:34:15Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.