Solving Oscillator Ordinary Differential Equations in the Time Domain with High Performance via Soft-constrained Physics-informed Neural Network with Small Data

• URL: http://arxiv.org/abs/2408.11077v5
• Date: Mon, 20 Oct 2025 15:41:38 GMT
• Title: Solving Oscillator Ordinary Differential Equations in the Time Domain with High Performance via Soft-constrained Physics-informed Neural Network with Small Data
• Authors: Kai-liang Lu,
• Abstract summary: 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.
• Score: 0.6446246430600296
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: In many scientific and engineering (e.g., physical, biochemical, medical) practices, data generated through expensive experiments or large-scale simulations, are often sparse and noisy. Physics-informed neural network (PINN) incorporates physical information and knowledge into network topology or computational processes as model priors, with the unique advantage of achieving strong generalization with small data. This study aims to investigate the performance characteristics of the soft-constrained PINN method to solving typical linear and nonlinear ordinary differential equations (ODEs) such as primer, Van der Pol and Duffing oscillators, especially the effectiveness, efficiency, and robustness to noise with minimal data. It is verified that the soft-constrained PINN significantly reduces the need for labeled data. With the aid of appropriate collocation points no need to be labeled, it can predict and also extrapolate with minimal data. First-order and second-order ODEs, no matter linear or nonlinear oscillators, require only one and two training data (containing initial values) respectively, just like classical analytic or Runge-Kutta methods, and with equivalent precision and comparable efficiency (fast training in seconds for scalar ODEs). Furthermore, it can conveniently impose a physical law (e.g., conservation of energy) constraint by adding a regularization term to the total loss function, improving the performance to deal with various complexities such as nonlinearity like Duffing. The DeepXDE-based PINN implementation is light code and can be efficiently trained on both GPU and CPU platforms. The mathematical and computational framework of this alternative and feasible PINN method to ODEs, can be easily extended to PDEs, etc., and is becoming a favorable catalyst for the era of Digital Twins.

Related papers

• Physics-Informed Laplace Neural Operator for Solving Partial Differential Equations [11.064132774859553]
  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.
  arXiv  Detail & Related papers  (2026-02-13T08:19:40Z)
• Data-Efficient Time-Dependent PDE Surrogates: Graph Neural Simulators vs. Neural Operators [0.0]
  We propose Neural Graph Simulators (GNS) as a principled surrogate modeling paradigm for time-dependent partial differential equations (PDEs)<n>GNS leverages message-passing combined with numerical time-stepping schemes to learn PDE dynamics by modeling the instantaneous time derivatives.<n>Results demonstrate that GNS is markedly more data-efficient, achieving less than 1% relative L2 error using only 3% of available trajectories.
  arXiv  Detail & Related papers  (2025-09-07T17:54:23Z)
• Spectral-inspired Neural Operator for Data-efficient PDE Simulation in Physics-agnostic Regimes [11.74789494197836]
  Partial equations (PDEs) govern evolution of various physical systems.<n>Classical numerical solvers require fine discretization and full knowledge of governing PDEs.<n>Data-driven neural PDE solvers mitigate these constraints by learning from data but demand large training.
  arXiv  Detail & Related papers  (2025-05-27T07:25:13Z)
• LaPON: A Lagrange's-mean-value-theorem-inspired operator network for solving PDEs and its application on NSE [8.014720523981385]
  We propose LaPON, an operator network inspired by the Lagrange's mean value theorem.<n>It embeds prior knowledge directly into the neural network architecture instead of the loss function.<n>LaPON provides a scalable and reliable solution for high-fidelity fluid dynamics simulation.
  arXiv  Detail & Related papers  (2025-05-18T10:45:17Z)
• Enabling Automatic Differentiation with Mollified Graph Neural Operators [73.52999622724101]
  We propose the mollified graph neural operator ($m$GNO), the first method to leverage automatic differentiation and compute exact gradients on arbitrary geometries.<n>For a PDE example on regular grids, $m$GNO 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)
• Method of data forward generation with partial differential equations for machine learning modeling in fluid mechanics [1.9688252014450927]
  This study proposes a high-efficient data forward generation method from the partial differential equations (PDEs)<n>A Poisson neural network (Poisson-NN) embedded in projection method and a wavelet transform convolutional neuro network (WTCNN) embedded in multigrid numerical simulation for solving incompressible Navier-Stokes equations is respectively proposed.
  arXiv  Detail & Related papers  (2025-01-06T15:17:13Z)
• Improving PINNs By Algebraic Inclusion of Boundary and Initial Conditions [0.1874930567916036]
  "AI for Science" aims to solve fundamental scientific problems using AI techniques. In this work we explore the possibility of changing the model being trained from being just a neural network to being a non-linear transformation of it. This reduces the number of terms in the loss function than the standard PINN losses.
  arXiv  Detail & Related papers  (2024-07-30T11:19:48Z)
• 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)
• Characteristics-Informed Neural Networks for Forward and Inverse Hyperbolic Problems [0.0]
  We propose characteristic-informed neural networks (CINN) for solving forward and inverse problems involving hyperbolic PDEs. CINN encodes the characteristics of the PDE in a general-purpose deep neural network trained with the usual MSE data-fitting regression loss. Preliminary results indicate that CINN is able to improve on the accuracy of the baseline PINN, while being nearly twice as fast to train and avoiding non-physical solutions.
  arXiv  Detail & Related papers  (2022-12-28T18:38:53Z)
• Physics-guided Data Augmentation for Learning the Solution Operator of Linear Differential Equations [2.1850269949775663]
  We propose a physics-guided data augmentation (PGDA) method to improve the accuracy and generalization of neural operator models. We demonstrate the advantage of PGDA on a variety of linear differential equations, showing that PGDA can improve the sample complexity and is robust to distributional shift.
  arXiv  Detail & Related papers  (2022-12-08T06:29:15Z)
• Reduced-PINN: An Integration-Based Physics-Informed Neural Networks for Stiff ODEs [0.0]
  Physics-informed neural networks (PINNs) have recently received much attention due to their capabilities in solving both forward and inverse problems. We propose a new PINN architecture, called Reduced-PINN, that utilizes a reduced-order integration method to enable the PINN to solve stiff chemical kinetics.
  arXiv  Detail & Related papers  (2022-08-23T09:20:42Z)
• Learning Physics-Informed Neural Networks without Stacked Back-propagation [82.26566759276105]
  We develop a novel approach that can significantly accelerate the training of Physics-Informed Neural Networks. In particular, we parameterize the PDE solution by the Gaussian smoothed model and show that, derived from Stein's Identity, the second-order derivatives can be efficiently calculated without back-propagation. Experimental results show that our proposed method can achieve competitive error compared to standard PINN training but is two orders of magnitude faster.
  arXiv  Detail & Related papers  (2022-02-18T18:07:54Z)
• 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)
• Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNs [4.560331122656578]
  Partial Differential Equations (PDEs) are notoriously difficult to solve. In this paper, we propose to approach the solution of PDEs based on a novel technique that combines the advantages of two recently emerging machine learning based approaches.
  arXiv  Detail & Related papers  (2021-09-15T08:10:23Z)
• Characterizing possible failure modes in physics-informed neural networks [55.83255669840384]
  Recent work in scientific machine learning has developed so-called physics-informed neural network (PINN) models. We demonstrate that, while existing PINN methodologies can learn good models for relatively trivial problems, they can easily fail to learn relevant physical phenomena even for simple PDEs. We show that these possible failure modes are not due to the lack of expressivity in the NN architecture, but that the PINN's setup makes the loss landscape very hard to optimize.
  arXiv  Detail & Related papers  (2021-09-02T16:06:45Z)
• Efficient training of physics-informed neural networks via importance sampling [2.9005223064604078]
  Physics-In Neural Networks (PINNs) are a class of deep neural networks that are trained to compute systems governed by partial differential equations (PDEs) We show that an importance sampling approach will improve the convergence behavior of PINNs training.
  arXiv  Detail & Related papers  (2021-04-26T02:45:10Z)
• dNNsolve: an efficient NN-based PDE solver [62.997667081978825]
  We introduce dNNsolve, that makes use of dual Neural Networks to solve ODEs/PDEs. We show that dNNsolve is capable of solving a broad range of ODEs/PDEs in 1, 2 and 3 spacetime dimensions.
  arXiv  Detail & Related papers  (2021-03-15T19:14:41Z)
• 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)
• 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)

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.