Physics-Informed Laplace Neural Operator for Solving Partial Differential Equations

• URL: http://arxiv.org/abs/2602.12706v1
• Date: Fri, 13 Feb 2026 08:19:40 GMT
• Title: Physics-Informed Laplace Neural Operator for Solving Partial Differential Equations
• Authors: Heechang Kim, Qianying Cao, Hyomin Shin, Seungchul Lee, George Em Karniadakis, Minseok Choi,
• Abstract summary: Physics-Informed Laplace Neural Operator (PILNO) is a fast surrogate solver for partial differential equations.<n>It embeds physics into training through PDE, boundary condition, and initial condition residuals.<n>PILNO consistently improves accuracy in small-data settings, reduces run-to-run variability across random seeds, and achieves stronger generalization than purely data-driven baselines.
• Score: 11.064132774859553
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs). However, purely data-driven models often require extensive training data and can generalize poorly, especially in small-data regimes and under unseen (out-of-distribution) input functions that are not represented in the training data. To address these limitations, we propose the Physics-Informed Laplace Neural Operator (PILNO), which enhances the Laplace Neural Operator (LNO) by embedding governing physics into training through PDE, boundary condition, and initial condition residuals. To improve expressivity, we first introduce an Advanced LNO (ALNO) backbone that retains a pole-residue transient representation while replacing the steady-state branch with an FNO-style Fourier multiplier. To make physics-informed training both data-efficient and robust, PILNO further leverages (i) virtual inputs: an unlabeled ensemble of input functions spanning a broad spectral range that provides abundant physics-only supervision and explicitly targets out-of-distribution (OOD) regimes; and (ii) temporal-causality weighting: a time-decaying reweighting of the physics residual that prioritizes early-time dynamics and stabilizes optimization for time-dependent PDEs. Across four representative benchmarks -- Burgers' equation, Darcy flow, a reaction-diffusion system, and a forced KdV equation -- PILNO consistently improves accuracy in small-data settings (e.g., N_train <= 27), reduces run-to-run variability across random seeds, and achieves stronger OOD generalization than purely data-driven baselines.

Related papers

• Accelerating PDE Solvers with Equation-Recast Neural Operator Preconditioning [9.178290601589365]
  Minimal-Data Parametric Neural Operator Preconditioning (MD-PNOP) is a new paradigm for accelerating parametric PDE solvers.<n>It recasts the residual from parameter deviation as additional source term, where trained neural operators can be used to refine the solution in an offline fashion.<n>It consistently achieves 50% reduction in computational time while maintaining full order fidelity for fixed-source, single-group eigenvalue, and multigroup coupled eigenvalue problems.
  arXiv  Detail & Related papers  (2025-09-01T12:14:58Z)
• PhysicsCorrect: A Training-Free Approach for Stable Neural PDE Simulations [4.7903561901859355]
  We present PhysicsCorrect, a training-free correction framework that enforces PDE consistency at each prediction step.<n>Our key innovation is an efficient caching strategy that precomputes the Jacobian and its pseudoinverse during an offline warm-up phase.<n>Across three representative PDE systems, PhysicsCorrect reduces prediction errors by up to 100x while adding negligible inference time.
  arXiv  Detail & Related papers  (2025-07-03T01:22:57Z)
• PMNO: A novel physics guided multi-step neural operator predictor for partial differential equations [23.04840527974364]
  We propose a novel physics guided multi-step neural operator (PMNO) architecture to address challenges in long-horizon prediction of complex physical systems.<n>The PMNO framework replaces the single-step input with multi-step historical data in the forward pass and introduces an implicit time-stepping scheme during backpropagation.<n>We demonstrate the superior predictive performance of PMNO predictor across a diverse range of physical systems.
  arXiv  Detail & Related papers  (2025-06-02T12:33:50Z)
• Solving Oscillator Ordinary Differential Equations in the Time Domain with High Performance via Soft-constrained Physics-informed Neural Network with Small Data [0.6446246430600296]
  Physics-informed neural network (PINN) incorporates physical information and knowledge into network topology or computational processes as model priors.<n>This study aims to investigate the performance characteristics of the soft-constrained PINN method to solve typical linear and nonlinear ordinary differential equations.
  arXiv  Detail & Related papers  (2024-08-19T13:02:06Z)
• DeltaPhi: Physical States Residual Learning for Neural Operators in Data-Limited PDE Solving [54.605760146540234]
  DeltaPhi is a novel learning framework that transforms the PDE solving task from learning direct input-output mappings to learning the residuals between similar physical states.<n>Extensive experiments demonstrate consistent and significant improvements across diverse physical systems.
  arXiv  Detail & Related papers  (2024-06-14T07:45:07Z)
• Guaranteed Approximation Bounds for Mixed-Precision Neural Operators [83.64404557466528]
  We build on intuition that neural operator learning inherently induces an approximation error. We show that our approach reduces GPU memory usage by up to 50% and improves throughput by 58% with little or no reduction in accuracy.
  arXiv  Detail & Related papers  (2023-07-27T17:42:06Z)
• Machine learning in and out of equilibrium [58.88325379746631]
  Our study uses a Fokker-Planck approach, adapted from statistical physics, to explore these parallels. We focus in particular on the stationary state of the system in the long-time limit, which in conventional SGD is out of equilibrium. We propose a new variation of Langevin dynamics (SGLD) that harnesses without replacement minibatching.
  arXiv  Detail & Related papers  (2023-06-06T09:12:49Z)
• Implicit Stochastic Gradient Descent for Training Physics-informed Neural Networks [51.92362217307946]
  Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems. PINNs are trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features. In this paper, we propose to employ implicit gradient descent (ISGD) method to train PINNs for improving the stability of training process.
  arXiv  Detail & Related papers  (2023-03-03T08:17:47Z)
• NeuralStagger: Accelerating Physics-constrained Neural PDE Solver with Spatial-temporal Decomposition [67.46012350241969]
  This paper proposes a general acceleration methodology called NeuralStagger. It decomposing the original learning tasks into several coarser-resolution subtasks. We demonstrate the successful application of NeuralStagger on 2D and 3D fluid dynamics simulations.
  arXiv  Detail & Related papers  (2023-02-20T19:36:52Z)
• FC-PINO: High Precision Physics-Informed Neural Operators via Fourier Continuation [60.706803227003995]
  We introduce the FC-PINO (Fourier-Continuation-based PINO) architecture which extends the accuracy and efficiency of PINO to non-periodic and non-smooth PDEs.<n>We demonstrate that standard PINO struggles to solve non-periodic and non-smooth PDEs with high precision, across challenging benchmarks.<n>In contrast, the proposed FC-PINO provides accurate, robust, and scalable solutions, substantially outperforming PINO alternatives.
  arXiv  Detail & Related papers  (2022-11-29T06:37:54Z)
• 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)
• Long-time integration of parametric evolution equations with physics-informed DeepONets [0.0]
  We introduce an effective framework for learning infinite-dimensional operators that map random initial conditions to associated PDE solutions within a short time interval. Global long-time predictions across a range of initial conditions can be then obtained by iteratively evaluating the trained model. This introduces a new approach to temporal domain decomposition that is shown to be effective in performing accurate long-time simulations.
  arXiv  Detail & Related papers  (2021-06-09T20:46:17Z)
• Incorporating NODE with Pre-trained Neural Differential Operator for Learning Dynamics [73.77459272878025]
  We propose to enhance the supervised signal in learning dynamics by pre-training a neural differential operator (NDO) NDO is pre-trained on a class of symbolic functions, and it learns the mapping between the trajectory samples of these functions to their derivatives. We provide theoretical guarantee on that the output of NDO can well approximate the ground truth derivatives by proper tuning the complexity of the library.
  arXiv  Detail & Related papers  (2021-06-08T08:04:47Z)

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.