Physics-Informed Neural Networks with Unknown Partial Differential Equations: an Application in Multivariate Time Series

2025-03-26

AI Summary

Accurate forecasting of multivariate time series governed by unknown partial differential equations (PDEs) remains challenging under sparse, noisy, or incomplete observational data. Method: We propose an end-to-end, data-driven framework that jointly discovers governing PDEs and builds physics-constrained models. Leveraging symbolic regression integrated with sparse identification of nonlinear dynamics (SINDy), we automatically infer PDE structure and embed it uniformly into three distinct modeling paradigms: physics-informed neural networks (PINNs), Bayesian PINNs, and Bayesian linear regression. Joint optimization of physics-informed losses and Bayesian objectives enables simultaneous inference of PDE parameters and predictive uncertainty. Contribution/Results: This work achieves the first seamless integration of automated PDE discovery with multi-paradigm physics-guided learning—requiring no prior knowledge of the underlying PDE. Experiments on real-world multivariate time-series datasets demonstrate substantial improvements in both forecast accuracy and uncertainty calibration, particularly under low-quality data conditions, outperforming purely data-driven baselines.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Interdisciplinary science discovery with web data mining

Abstract

A significant advancement in Neural Network (NN) research is the integration of domain-specific knowledge through custom loss functions. This approach addresses a crucial challenge: how can models utilize physics or mathematical principles to enhance predictions when dealing with sparse, noisy, or incomplete data? Physics-Informed Neural Networks (PINNs) put this idea into practice by incorporating physical equations, such as Partial Differential Equations (PDEs), as soft constraints. This guidance helps the networks find solutions that align with established laws. Recently, researchers have expanded this framework to include Bayesian NNs (BNNs), which allow for uncertainty quantification while still adhering to physical principles. But what happens when the governing equations of a system are not known? In this work, we introduce methods to automatically extract PDEs from historical data. We then integrate these learned equations into three different modeling approaches: PINNs, Bayesian-PINNs (B-PINNs), and Bayesian Linear Regression (BLR). To assess these frameworks, we evaluate them on a real-world Multivariate Time Series (MTS) dataset. We compare their effectiveness in forecasting future states under different scenarios: with and without PDE constraints and accuracy considerations. This research aims to bridge the gap between data-driven discovery and physics-guided learning, providing valuable insights for practical applications.

Problem

• Extracting unknown PDEs from historical data automatically
• Integrating learned PDEs into PINNs, B-PINNs, and BLR models
• Evaluating physics-guided forecasting in multivariate time series

Innovation

• Automatically extract PDEs from historical data
• Integrate learned PDEs into PINNs and B-PINNs
• Combine Bayesian methods with physics-guided learning