Deep Semiparametric Partial Differential Equation Models

2025-06-28

AI Summary

For dynamic systems with partially known physical mechanisms but challenging-to-model microscopic effects, this paper proposes a semiparametric partial differential equation (PDE) framework that jointly incorporates prior physical laws and data-driven nonparametric components. Methodologically, we design a deep profile M-estimation strategy that decouples PDE solving from parametric and nonparametric learning, thereby integrating numerical PDE accuracy with neural network expressivity. Theoretically, we establish the first inference theory for semiparametric PDEs under deep M-estimation, characterizing how PDE structure governs the convergence rate of the nonparametric component—enhancing both interpretability and parameter efficiency. Experiments on synthetic and real-world data demonstrate accurate identification of underlying dynamical mechanisms while maintaining high predictive accuracy and mechanistic interpretability, offering a novel paradigm for scientific machine learning.

Technology Category

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

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

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.

Problem

• Modeling data dynamics with interpretable and flexible PDE components
• Decoupling PDE solutions using deep M-estimation for accurate inference
• Balancing physical laws and data-driven approaches for complex systems

Innovation

• Combines physical laws with data-driven components
• Uses deep profiling M-estimation for PDE solutions
• Establishes semiparametric inference for deep M-estimation