CompNO: A Novel Foundation Model approach for solving Partial Differential Equations

• URL: http://arxiv.org/abs/2601.07384v1
• Date: Mon, 12 Jan 2026 10:04:48 GMT
• Title: CompNO: A Novel Foundation Model approach for solving Partial Differential Equations
• Authors: Hamda Hmida, Hsiu-Wen Chang Joly, Youssef Mesri,
• Abstract summary: Partial differential equations govern a wide range of physical phenomena, but their numerical solution remains computationally demanding.<n>Recent Scientific Foundation Models (SFMs) aim to alleviate this cost by learning universal surrogates from large collections of simulated systems.<n>We introduce Compositional Neural Operators (CompNO), a compositional neural operator framework for parametric PDEs.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Partial differential equations (PDEs) govern a wide range of physical phenomena, but their numerical solution remains computationally demanding, especially when repeated simulations are required across many parameter settings. Recent Scientific Foundation Models (SFMs) aim to alleviate this cost by learning universal surrogates from large collections of simulated systems, yet they typically rely on monolithic architectures with limited interpretability and high pretraining expense. In this work we introduce Compositional Neural Operators (CompNO), a compositional neural operator framework for parametric PDEs. Instead of pretraining a single large model on heterogeneous data, CompNO first learns a library of Foundation Blocks, where each block is a parametric Fourier neural operator specialized to a fundamental differential operator (e.g. convection, diffusion, nonlinear convection). These blocks are then assembled, via lightweight Adaptation Blocks, into task-specific solvers that approximate the temporal evolution operator for target PDEs. A dedicated boundary-condition operator further enforces Dirichlet constraints exactly at inference time. We validate CompNO on one-dimensional convection, diffusion, convection--diffusion and Burgers' equations from the PDEBench suite. The proposed framework achieves lower relative L2 error than strong baselines (PFNO, PDEFormer and in-context learning based models) on linear parametric systems, while remaining competitive on nonlinear Burgers' flows. The model maintains exact boundary satisfaction with zero loss at domain boundaries, and exhibits robust generalization across a broad range of Peclet and Reynolds numbers. These results demonstrate that compositional neural operators provide a scalable and physically interpretable pathway towards foundation models for PDEs.

Related papers

• Discontinuous Galerkin finite element operator network for solving non-smooth PDEs [15.286345729268149]
  We introduce Discontinuous Galerkin Finite Element Operator Network (DG--FEONet), a data-free operator learning framework.<n>It combines the strengths of the discontinuous Galerkin (DG) method with neural networks to solve parametric partial differential equations.<n>Our results highlight the potential of combining local discretization schemes with machine learning to achieve robust, singularity-aware operator approximation.
  arXiv  Detail & Related papers  (2026-01-07T07:43:30Z)
• Expanding the Chaos: Neural Operator for Stochastic (Partial) Differential Equations [65.80144621950981]
  We build on Wiener chaos expansions (WCE) to design neural operator (NO) architectures for SPDEs and SDEs.<n>We show that WCE-based neural operators provide a practical and scalable way to learn SDE/SPDE solution operators.
  arXiv  Detail & Related papers  (2026-01-03T00:59:25Z)
• Physics-informed low-rank neural operators with application to parametric elliptic PDEs [0.0]
  We present PILNO, a neural operator framework for approximating solution operators of partial differential equations (PDEs) on point cloud data.<n>PILNO combines low-rank kernel approximations with an encoder--decoder architecture, enabling fast, continuous one-shot predictions while remaining independent of specific discretizations.<n>We demonstrate its effectiveness on diverse problems, including function fitting, the Poisson equation, the screened Poisson equation with variable coefficients, and parameterized Darcy flow.
  arXiv  Detail & Related papers  (2025-09-09T12:54:06Z)
• Scale-Consistent Learning for Partial Differential Equations [79.48661503591943]
  We propose a data augmentation scheme based on scale-consistency properties of PDEs.<n>We then design a scale-informed neural operator that can model a wide range of scales.<n>With scale-consistency, the model trained on $Re$ of 1000 can generalize to $Re$ ranging from 250 to 10000.
  arXiv  Detail & Related papers  (2025-07-24T21:29:52Z)
• Neural Expectation Operators [2.1756081703276]
  This paper introduces textbfMeasure Learning, a paradigm for modeling ambiguity via non-linear expectations.<n>We define Neural Expectation Operators as solutions to Backward Differential Equations (BSDEs) whose drivers are parameterized by neural networks.<n>We provide constructive methods for enforcing key axiomatic properties, such as convexity, by architectural design.
  arXiv  Detail & Related papers  (2025-07-13T06:19:28Z)
• Spectral-inspired Operator Learning with Limited Data and Unknown Physics [10.143396024546368]
  Spectral-Inspired Neural Operator (SINO) can model complex systems from just 2-5 trajectories without requiring explicit PDE terms.<n>To model nonlinear effects, SINO employs a Pi-block that performs multiplicative operations on spectral features, complemented by a low-pass filter to suppress aliasing.<n>Experiments on both 2D and 3D PDE benchmarks demonstrate that SINO achieves state-of-the-art performance, with improvements of 1-2 orders of magnitude in accuracy.
  arXiv  Detail & Related papers  (2025-05-27T07:25:13Z)
• Physics-informed reduced order model with conditional neural fields [4.5355909674008865]
  This study presents the conditional neural fields for reduced-order modeling (CNF-ROM) framework to approximate solutions of parametrized partial differential equations (PDEs)<n>The approach combines a parametric neural ODE for modeling latent dynamics over time with a decoder that reconstructs PDE solutions from the corresponding latent states.
  arXiv  Detail & Related papers  (2024-12-06T18:04:33Z)
• 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)
• LordNet: An Efficient Neural Network for Learning to Solve Parametric Partial Differential Equations without Simulated Data [47.49194807524502]
  We propose LordNet, a tunable and efficient neural network for modeling entanglements. The experiments on solving Poisson's equation and (2D and 3D) Navier-Stokes equation demonstrate that the long-range entanglements can be well modeled by the LordNet.
  arXiv  Detail & Related papers  (2022-06-19T14:41:08Z)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)
• 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)
• Large-scale Neural Solvers for Partial Differential Equations [48.7576911714538]
  Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. Recent numerical solvers require manual discretization of the underlying equation as well as sophisticated, tailored code for distributed computing. We examine the applicability of continuous, mesh-free neural solvers for partial differential equations, physics-informed neural networks (PINNs) We discuss the accuracy of GatedPINN with respect to analytical solutions -- as well as state-of-the-art numerical solvers, such as spectral solvers.
  arXiv  Detail & Related papers  (2020-09-08T13:26:51Z)

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.