Deep Semiparametric Partial Differential Equation Models

TL;DR

This paper introduces a semiparametric PDE modeling framework combining physical laws with neural networks, enabling interpretable and adaptive analysis of complex data dynamics, supported by new inference theory and demonstrated through simulations and real data.

Contribution

It proposes a novel deep semiparametric PDE model with a deep profiling M-estimation approach, integrating physical mechanisms and data-driven components, along with establishing inference theory for such models.

Findings

The method effectively captures complex data dynamics.

Theoretical analysis shows convergence and efficiency.

Empirical results validate the model's interpretability and accuracy.

Abstract

In many scientific fields, the generation and evolution of data are governed by partial differential equations (PDEs) which are typically informed by established physical laws at the macroscopic level to describe general and predictable dynamics. However, some complex influences may not be fully captured by these laws at the microscopic level due to limited scientific understanding. This work proposes a unified framework to model, estimate, and infer the mechanisms underlying data dynamics. We introduce a general semiparametric PDE (SemiPDE) model that combines interpretable mechanisms based on physical laws with flexible data-driven components to account for unknown effects. The physical mechanisms enhance the SemiPDE model's stability and interpretability, while the data-driven components improve adaptivity to complex real-world scenarios. A deep profiling M-estimation approach is proposed to decouple the solutions of PDEs in the estimation procedure, leveraging both the accuracy of numerical methods for solving PDEs and the expressive power of neural networks. For the first time, we establish a semiparametric inference method and theory for deep M-estimation, considering both training dynamics and complex PDE models. We analyze how the PDE structure affects the convergence rate of the nonparametric estimator, and consequently, the parametric efficiency and inference procedure enable the identification of interpretable mechanisms governing data dynamics. Simulated and real-world examples demonstrate the effectiveness of the proposed methodology and support the theoretical findings.