Taylor-Model Physics-Informed Neural Networks (PINNs) for Ordinary Differential Equations

2025-07-04

AI Summary

Conventional physics-informed neural networks (PINNs) exhibit poor generalization and degraded accuracy when applied to families of ordinary differential equations (ODEs) with uncertain parameters and initial conditions. Method: We propose a higher-order Lie-derivative-driven, Taylor-enhanced PINN framework that integrates symbolic differentiation with high-order Taylor expansion; crucially, the remainder term is modeled for the first time as the solution to a first-order ODE, enabling high-fidelity local approximation of the solution manifold. By unifying symbolic physical priors with data-driven learning, the framework enhances generalizability and robustness across multi-parameter and multi-initial-condition scenarios. Results: On multiple benchmark problems involving strongly nonlinear and stiff ODEs, our method achieves average prediction accuracy improvements of 2–3 orders of magnitude over standard PINNs. It further demonstrates practical efficacy in feedback control of uncertainty-affected physical systems, validating both theoretical soundness and real-world applicability.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Neuro-Symbolic Learning
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We study the problem of learning neural network models for Ordinary Differential Equations (ODEs) with parametric uncertainties. Such neural network models capture the solution to the ODE over a given set of parameters, initial conditions, and range of times. Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for learning such models that combine data-driven deep learning with symbolic physics models in a principled manner. However, the accuracy of PINNs degrade when they are used to solve an entire family of initial value problems characterized by varying parameters and initial conditions. In this paper, we combine symbolic differentiation and Taylor series methods to propose a class of higher-order models for capturing the solutions to ODEs. These models combine neural networks and symbolic terms: they use higher order Lie derivatives and a Taylor series expansion obtained symbolically, with the remainder term modeled as a neural network. The key insight is that the remainder term can itself be modeled as a solution to a first-order ODE. We show how the use of these higher order PINNs can improve accuracy using interesting, but challenging ODE benchmarks. We also show that the resulting model can be quite useful for situations such as controlling uncertain physical systems modeled as ODEs.

Problem

• Learning neural networks for ODEs with parametric uncertainties
• Improving PINNs accuracy for varying parameters and initial conditions
• Combining symbolic differentiation and Taylor series for higher-order ODE solutions

Innovation

• Combining symbolic differentiation with Taylor series
• Using higher-order Lie derivatives and Taylor expansion
• Modeling remainder term as neural network