Related papers: One-Shot Transfer Learning for Nonlinear ODEs

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Chebyshev-Augmented One-Shot Transfer Learning for PINNs on Nonlinear Differential Equations [0.794957965474334]
Physics-Informed Neural Networks (PINNs) offer a flexible paradigm for solving differential equations by governing governing laws into the network.<n>A limitation is instance specificity: standard PINNs typically require retraining for each new forcing term, latent/initial condition, or nonlinear problems closed-form adaptation.<n>In this paper we address the class of nonlinearities amenable to one-shot PINN transfer by combining OTL with Chebyshev surrogates.
arXiv Detail & Related papers (2026-05-02T22:49:37Z)
PTL-PINNs: Perturbation-Guided Transfer Learning with Physics- Informed Neural Networks for Nonlinear Systems [7.961515776672606]
We propose a perturbation-guided transfer learning framework for PINNs (PTL-PINN)<n>PTL-PINNs solve an approximate linear perturbative system using closed-form expressions, enabling rapid generalization with the time complexity of matrix-vector multiplication.<n>We show that PTL-PINNs achieve accuracy comparable to various Runge-Kutta methods, with computational speeds up to one order of magnitude faster.
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arXiv Detail & Related papers (2026-01-15T14:43:21Z)
One-Shot Transfer Learning for Nonlinear PDEs with Perturbative PINNs [0.794957965474334]
We propose a framework for solving nonlinear partial differential equations.<n>Our contributions are: (i) extending one-shot transfer learning from nonlinear ODEs to PDEs, (ii) deriving a closed-form solution for adapting to new PDE instances, and (iii) demonstrating accuracy and efficiency on canonical nonlinear PDEs.
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A modal decomposition leads to a coupled nonlinear system of ordinary differential equations.<n>Recent work in applied machine learning approaches has been used to model lumped dynamic systems automatically from data.<n>We show that the model can be trained to reproduce the nonlinear dynamics of the system.
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We propose a novel approach for training Physics-enhanced Neural ODEs (PeN-ODEs) by expressing the training process as a dynamic optimization problem.<n>The full model, including neural components, is discretized using a high-order implicit Runge-Kutta method with flipped Legendre-Gauss-Radau points.<n>This formulation enables simultaneous optimization of network parameters and state trajectories, addressing key limitations of ODE solver-based training in terms of stability, runtime, and accuracy.
arXiv Detail & Related papers (2025-05-06T14:04:46Z)
Efficient PINNs via Multi-Head Unimodular Regularization of the Solutions Space [0.6474760227870044]
We present a machine learning framework to facilitate the solution of nonlinear multiscale differential equations.<n>This framework is based on what is called textitmulti-head (MH) training, which involves training the network to learn a general space of all solutions.<n>We show that the multi-head approach, combined with Unimodular Regularization, significantly improves the efficiency of PINNs.
arXiv Detail & Related papers (2025-01-21T13:25:56Z)
Physics-informed nonlinear vector autoregressive models for the prediction of dynamical systems [0.36248657646376703]
We focus on one class of models called nonlinear vector autoregression (N VAR) to solve ordinary differential equations (ODEs) Motivated by connections to numerical integration and physics-informed neural networks, we explicitly derive the physics-informed N VAR. Because N VAR and piN VAR completely share their learned parameters, we propose an augmented procedure to jointly train the two models. We evaluate the ability of the piN VAR model to predict solutions to various ODE systems, such as the undamped spring, a Lotka-Volterra predator-prey nonlinear model, and the chaotic Lorenz system.
arXiv Detail & Related papers (2024-07-25T14:10:42Z)
Newton Informed Neural Operator for Computing Multiple Solutions of Nonlinear Partials Differential Equations [3.8916312075738273]
We propose a novel approach called the Newton Informed Neural Operator to tackle nonlinearities. Our method combines classical Newton methods, addressing well-posed problems, and efficiently learns multiple solutions in a single learning process.
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A Stable and Scalable Method for Solving Initial Value PDEs with Neural Networks [52.5899851000193]
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arXiv Detail & Related papers (2023-04-28T17:28:18Z)
Learning Neural Constitutive Laws From Motion Observations for Generalizable PDE Dynamics [97.38308257547186]
Many NN approaches learn an end-to-end model that implicitly models both the governing PDE and material models. We argue that the governing PDEs are often well-known and should be explicitly enforced rather than learned. We introduce a new framework termed "Neural Constitutive Laws" (NCLaw) which utilizes a network architecture that strictly guarantees standard priors.
arXiv Detail & Related papers (2023-04-27T17:42:24Z)
Learning Subgrid-scale Models with Neural Ordinary Differential Equations [0.39160947065896795]
We propose a new approach to learning the subgrid-scale model when simulating partial differential equations (PDEs) In this approach neural networks are used to learn the coarse- to fine-grid map, which can be viewed as subgrid-scale parameterization. Our method inherits the advantages of NODEs and can be used to parameterize subgrid scales, approximate coupling operators, and improve the efficiency of low-order solvers.
arXiv Detail & Related papers (2022-12-20T02:45:09Z)
Experimental study of Neural ODE training with adaptive solver for dynamical systems modeling [72.84259710412293]
Some ODE solvers called adaptive can adapt their evaluation strategy depending on the complexity of the problem at hand. This paper describes a simple set of experiments to show why adaptive solvers cannot be seamlessly leveraged as a black-box for dynamical systems modelling.
arXiv Detail & Related papers (2022-11-13T17:48:04Z)
NeuralPDE: Modelling Dynamical Systems from Data [0.44259821861543996]
We propose NeuralPDE, a model which combines convolutional neural networks (CNNs) with differentiable ODE solvers to model dynamical systems. We show that the Method of Lines used in standard PDE solvers can be represented using convolutions which makes CNNs the natural choice to parametrize arbitrary PDE dynamics. Our model can be applied to any data without requiring any prior knowledge about the governing PDE.
arXiv Detail & Related papers (2021-11-15T10:59:52Z)
Continuous Convolutional Neural Networks: Coupled Neural PDE and ODE [1.1897857181479061]
This work proposes a variant of Convolutional Neural Networks (CNNs) that can learn the hidden dynamics of a physical system. Instead of considering the physical system such as image, time -series as a system of multiple layers, this new technique can model a system in the form of Differential Equation (DEs)
arXiv Detail & Related papers (2021-10-30T21:45:00Z)
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)
Interpolation Technique to Speed Up Gradients Propagation in Neural ODEs [71.26657499537366]
We propose a simple literature-based method for the efficient approximation of gradients in neural ODE models. We compare it with the reverse dynamic method to train neural ODEs on classification, density estimation, and inference approximation tasks.
arXiv Detail & Related papers (2020-03-11T13:15:57Z)

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