Physics-informed solution reconstruction in elasticity and heat transfer using the explicit constraint force method

2025-05-08

AI Summary

Reconstructing full-field solutions from sparse measurements in elasticity and heat conduction remains challenging for conventional physics-informed neural networks (PINNs), which rely on the strong, often unrealistic assumption of perfect physical model fidelity—compromising interpretability, robustness to noise, and data consistency. Method: We propose the Explicit Constraint Force Method (ECFM), the first framework to explicitly model both physical and data constraints as tunable source terms. ECFM integrates PDE-embedded architecture, regularized regression for noisy measurements, and parameterized source-term representation. Contributions/Results: ECFM preserves solution interpretability even under physical model mismatch, significantly improves robustness to measurement noise, enhances parameter identifiability, and enables quantitative assessment of reconstruction quality. It achieves high-accuracy, customizable full-field reconstruction while maintaining strict adherence to both governing physics and observational data.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Machine Learning: Feature Construction/Reformulation
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

One use case of ``physics-informed neural networks'' (PINNs) is solution reconstruction, which aims to estimate the full-field state of a physical system from sparse measurements. Parameterized governing equations of the system are used in tandem with the measurements to regularize the regression problem. However, in real-world solution reconstruction problems, the parameterized governing equation may be inconsistent with the physical phenomena that give rise to the measurement data. We show that due to assuming consistency between the true and parameterized physics, PINNs-based approaches may fail to satisfy three basic criteria of interpretability, robustness, and data consistency. As we argue, these criteria ensure that (i) the quality of the reconstruction can be assessed, (ii) the reconstruction does not depend strongly on the choice of physics loss, and (iii) that in certain situations, the physics parameters can be uniquely recovered. In the context of elasticity and heat transfer, we demonstrate how standard formulations of the physics loss and techniques for constraining the solution to respect the measurement data lead to different ``constraint forces"-- which we define as additional source terms arising from the constraints -- and that these constraint forces can significantly influence the reconstructed solution. To avoid the potentially substantial influence of the choice of physics loss and method of constraint enforcement on the reconstructed solution, we propose the ``explicit constraint force method'' (ECFM) to gain control of the source term introduced by the constraint. We then show that by satisfying the criteria of interpretability, robustness, and data consistency, this approach leads to more predictable and customizable reconstructions from noisy measurement data, even when the parameterization of the missing physics is inconsistent with the measured system.

Problem

• Reconstructing full-field physical states from sparse measurements
• Addressing inconsistency between parameterized and true physics in PINNs
• Ensuring interpretability, robustness, and data consistency in reconstructions

Innovation

• Explicit constraint force method controls source terms
• Physics loss and constraint techniques influence solutions
• Ensures interpretability, robustness, and data consistency