Strategies for training point distributions in physics-informed neural networks

• URL: http://arxiv.org/abs/2508.13216v1
• Date: Sun, 17 Aug 2025 09:40:49 GMT
• Title: Strategies for training point distributions in physics-informed neural networks
• Authors: Santosh Humagain, Toni Schneidereit,
• Abstract summary: Physics-informed neural networks approach the approximation of differential equations by directly incorporating their structure and given conditions in a loss function.<n>In this paper, we investigate and evaluate a core component of the approach, namely the training point distribution.<n>The results show the impact of the training point on the solution accuracy and we find evidence that they are connected to the characteristics of the differential equation.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Physics-informed neural networks approach the approximation of differential equations by directly incorporating their structure and given conditions in a loss function. This enables conditions like, e.g., invariants to be easily added during the modelling phase. In addition, the approach can be considered as mesh free and can be utilised to compute solutions on arbitrary grids after the training phase. Therefore, physics-informed neural networks are emerging as a promising alternative to solving differential equations with methods from numerical mathematics. However, their performance highly depends on a large variety of factors. In this paper, we systematically investigate and evaluate a core component of the approach, namely the training point distribution. We test two ordinary and two partial differential equations with five strategies for training data generation and shallow network architectures, with one and two hidden layers. In addition to common distributions, we introduce sine-based training points, which are motivated by the construction of Chebyshev nodes. The results are challenged by using certain parameter combinations like, e.g., random and fixed-seed weight initialisation for reproducibility. The results show the impact of the training point distributions on the solution accuracy and we find evidence that they are connected to the characteristics of the differential equation.

Related papers

• DInf-Grid: A Neural Differential Equation Solver with Differentiable Feature Grids [73.28614344779076]
  We present a differentiable grid-based representation for efficiently solving differential equations (DEs)<n>Our results demonstrate a 5-20x speed-up over coordinate-based methods, solving differential equations in seconds or minutes while maintaining comparable accuracy and compactness.
  arXiv  Detail & Related papers  (2026-01-15T18:59:57Z)
• Self-Supervised Coarsening of Unstructured Grid with Automatic Differentiation [55.88862563823878]
  In this work, we present an original algorithm to coarsen an unstructured grid based on the concepts of differentiable physics.<n>We demonstrate performance of the algorithm on two PDEs: a linear equation which governs slightly compressible fluid flow in porous media and the wave equation.<n>Our results show that in the considered scenarios, we reduced the number of grid points up to 10 times while preserving the modeled variable dynamics in the points of interest.
  arXiv  Detail & Related papers  (2025-07-24T11:02:13Z)
• Enabling Automatic Differentiation with Mollified Graph Neural Operators [75.3183193262225]
  We propose the mollified graph neural operator (mGNO), the first method to leverage automatic differentiation and compute emphexact gradients on arbitrary geometries.<n>For a PDE example on regular grids, mGNO paired with autograd reduced the L2 relative data error by 20x compared to finite differences.<n>It can also solve PDEs on unstructured point clouds seamlessly, using physics losses only, at resolutions vastly lower than those needed for finite differences to be accurate enough.
  arXiv  Detail & Related papers  (2025-04-11T06:16:30Z)
• Solving partial differential equations with sampled neural networks [1.8590821261905535]
  Approximation of solutions to partial differential equations (PDE) is an important problem in computational science and engineering. We discuss how sampling the hidden weights and biases of the ansatz network from data-agnostic and data-dependent probability distributions allows us to progress on both challenges.
  arXiv  Detail & Related papers  (2024-05-31T14:24:39Z)
• Automatic Differentiation is Essential in Training Neural Networks for Solving Differential Equations [7.890817997914349]
  Neural network-based approaches have recently shown significant promise in solving partial differential equations (PDEs) in science and engineering.<n>One advantage of the neural network methods for PDEs lies in its automatic differentiation (AD)<n>In this paper, we quantitatively demonstrate the advantage of AD in training neural networks.
  arXiv  Detail & Related papers  (2024-05-23T02:01:05Z)
• Neural Physics: Using AI Libraries to Develop Physics-Based Solvers for Incompressible Computational Fluid Dynamics [2.6661542645011056]
  We propose to implement numerical discretisations as convolutional layers of a neural network.<n>We demonstrate that these systems can be solved entirely by functions in AI libraries.
  arXiv  Detail & Related papers  (2024-02-27T22:00:50Z)
• Physics-Informed Generator-Encoder Adversarial Networks with Latent Space Matching for Stochastic Differential Equations [14.999611448900822]
  We propose a new class of physics-informed neural networks to address the challenges posed by forward, inverse, and mixed problems in differential equations. Our model consists of two key components: the generator and the encoder, both updated alternately by gradient descent. In contrast to previous approaches, we employ an indirect matching that operates within the lower-dimensional latent feature space.
  arXiv  Detail & Related papers  (2023-11-03T04:29:49Z)
• Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations [64.78260098263489]
  In this work, we assess the ability of physics-informed neural networks (PINNs) to solve increasingly-complex coupled ordinary differential equations (ODEs) We show that PINNs eventually fail to produce correct solutions to these benchmarks as their complexity increases. We identify several reasons why this may be the case, including insufficient network capacity, poor conditioning of the ODEs, and high local curvature, as measured by the Laplacian of the PINN loss.
  arXiv  Detail & Related papers  (2022-10-14T15:01:32Z)
• Conditional physics informed neural networks [85.48030573849712]
  We introduce conditional PINNs (physics informed neural networks) for estimating the solution of classes of eigenvalue problems. We show that a single deep neural network can learn the solution of partial differential equations for an entire class of problems.
  arXiv  Detail & Related papers  (2021-04-06T18:29:14Z)
• Learning Neural Network Subspaces [74.44457651546728]
  Recent observations have advanced our understanding of the neural network optimization landscape. With a similar computational cost as training one model, we learn lines, curves, and simplexes of high-accuracy neural networks. With a similar computational cost as training one model, we learn lines, curves, and simplexes of high-accuracy neural networks.
  arXiv  Detail & Related papers  (2021-02-20T23:26:58Z)
• Unsupervised Learning of Solutions to Differential Equations with Generative Adversarial Networks [1.1470070927586016]
  We develop a novel method for solving differential equations with unsupervised neural networks. We show that our method, which we call Differential Equation GAN (DEQGAN), can obtain multiple orders of magnitude lower mean squared errors.
  arXiv  Detail & Related papers  (2020-07-21T23:36:36Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)
• Finite Difference Neural Networks: Fast Prediction of Partial Differential Equations [5.575293536755126]
  We propose a novel neural network framework, finite difference neural networks (FDNet), to learn partial differential equations from data. Specifically, our proposed finite difference inspired network is designed to learn the underlying governing partial differential equations from trajectory data.
  arXiv  Detail & Related papers  (2020-06-02T19:17:58Z)

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.