Long-time integration of parametric evolution equations with physics-informed DeepONets

TL;DR

The paper tackles the challenge of long-time prediction for parametric evolution equations by learning a short-time solution operator $G_{\theta}$ via physics-informed DeepONets that map initial-condition functions $u(\cdot)$ to solutions $s(\mathbf{x},t)$ over $t\in[0,\Delta t]$.Long-time predictions are then constructed through iterative application of the operator, effectively decomposing the temporal domain into manageable segments while leveraging physical constraints through a PDE residual-based loss.Across a suite of ODEs and PDEs, including inhomogeneous, stiff, wave, diffusion–reaction, and KdV systems, the approach yields accurate predictions with low relative $L^2$ errors (often sub-1%) and substantial speedups (10x–50x) over traditional solvers, while remaining data-efficient, particularly in the data–physics integrated KdV setting.The work demonstrates a practical ML-based pathway for scalable, high-fidelity scientific computation, with ongoing questions about stability, chaotic dynamics, and theoretical guarantees.Overall, physics-informed DeepONets offer a promising framework for rapid, long-horizon emulation of complex non-linear evolution equations.

Abstract

Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to introduce new effective ways of simulating PDEs, however existing approaches are not able to reliably return stable and accurate predictions across long temporal horizons. We aim to address this challenge by introducing an effective framework for learning infinite-dimensional operators that map random initial conditions to associated PDE solutions within a short time interval. Such latent operators can be parametrized by deep neural networks that are trained in an entirely self-supervised manner without requiring any paired input-output observations. Global long-time predictions across a range of initial conditions can be then obtained by iteratively evaluating the trained model using each prediction as the initial condition for the next evaluation step. This introduces a new approach to temporal domain decomposition that is shown to be effective in performing accurate long-time simulations for a wide range of parametric ODE and PDE systems, from wave propagation, to reaction-diffusion dynamics and stiff chemical kinetics, all at a fraction of the computational cost needed by classical numerical solvers.

Figures (22)

• Figure 1: Gravity Pendulum: Predicted solution for $s_1(t)$ and $s_2(t)$ versus the corresponding reference solution. The result is obtained by training a conventional PINN (5 layers, 100 hidden units, tanh activations) for $10^5$ iterations of gradient descent using the Adam optimizer. Evidently, the model predictions collapse after $t=10$.
• Figure 2: Making DeepOnets physics-informed: The DeepONet architecture lu2021learning consists of two sub-networks, the branch net for extracting latent representations of input functions, and the trunk net for extracting latent representations of input coordinates at which the output functions are evaluated. A continuous and differentiable representation of the output functions is then obtained by merging the latent representations extracted by each sub-network via a dot product. Automatic differentiation can then be employed to formulate appropriate regularization mechanisms for biasing the DeepOnet outputs to satisfy a given system of PDEs.
• Figure 3: Gravity Pendulum: Predicted solution for $s_1(t)$ and $s_2(t)$ versus the corresponding reference solution. The result is obtained by applying Algorithm \ref{['alg: long_time_integration']} to a trained a physics-informed DeepONet. The relative $L^2$ errors of $s_1$ and $s_2$ are $1.72\%$ and $1.63\%$, respectively.
• Figure 4: Inhomogeneous ODE: Exact solution versus the predicted solution of a trained physics-informed DeepONet using Algorithm \ref{['alg: long_time_integration']} to integrate the ODE system of equations (\ref{['eq: ODE_sin']}) - (\ref{['eq: ODE_sin_IC']}) up to $T = 1000$. The relative $L^2$ error is $0.84\%$.
• Figure 5: linear ODE: Relative $L^2$ prediction error of a physics-informed DeepONet as a function of the final time $T$, averaged over 100 different examples in the test data-set.