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

TL;DR

This work tackles the problem of physics-informed learning when governing PDEs are unknown by proposing a data-driven PDE extraction pipeline and integrating the inferred equations into PINN, B-PINN, and BLR frameworks. It demonstrates that surrogate neural networks, especially TCNs, can recover meaningful local PDEs from multivariate time-series data and that incorporating these PDEs into learning improves forecasting accuracy for several outputs. Bayesian variants (B-PINN, PI-BLR) offer uncertainty quantification but require more data and computation, revealing a trade-off between interpretability, robustness, and efficiency. The results suggest a practical pathway for applying physics-guided learning in domains lacking explicit governing equations, such as economics or social sciences, by coupling data-driven PDE discovery with physics-informed modeling.

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.

Figures (8)

• Figure 1: Predictions of the state variables in PDE extraction phase.
• Figure 2: Continues.
• Figure 3: Predictions of the derivatives.
• Figure 4: Continues.
• Figure 5: Continues.