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

TL;DR

This work tackles solution reconstruction under misspecified physics by exposing how standard PINN losses impose constraint forces that can compromise interpretability, robustness, and data consistency. It analyzes penalty, hard-constrained, weak-form, and energy-based formulations, revealing that constraint forces mediated by the chosen physics loss drive reconstructions in ways that may be unpredictable. The authors introduce the Explicit Constraint Force Method (ECFM), which inserts explicit spatial constraint forces at measurement locations and optimizes physics parameters to minimize the total constraint force, thereby achieving data-consistent, interpretable reconstructions that are robust to the form of the physics loss. Through 1D elastic, 1D hyperelastic, and 2D heat conduction examples, ECFM demonstrates controllable, transferable reconstructions and provides a physically meaningful metric—the total constraint force—for assessing parameter consistency and guiding model refinement. The approach offers a principled path to reconcile data and physics in elasticity and heat transfer, with potential extensions to broader dynamical systems and time-dependent constraints.

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.

Figures (15)

• Figure 1: Different techniques rely on different amounts of data and knowledge of the system's physics. Our explicit constraint force method (ECFM) does not rely on the assumption that the parameterized model is consistent with measurement data.
• Figure 2: Measurement data for the 1D linearly elastic BVP (left) and interpolation of the measurement data with the assumed body force model and penalty parameter (right). It is not clear whether the constraints are enforced as a result of recovering the physics parameters or from the penalty term.
• Figure 3: In this example, minimizing the energy $(\hat{\Pi})$ over $\epsilon$ will lead to poor interpolations (left), whereas the minimum strong form loss (which we call $Z$, to distinguish it from the loss involving the strong form and penalty in Eq. \ref{['L1']}), is obtained when the physics parameter is correct, leading to good interpolations (right). It is interesting to note that energy and strong form loss produce very different interpolations on what appears to be the same problem.
• Figure 4: Remedying the issues with the energy objective by using the minimum constraint force method (left). The minimum constraint force interpolation now accurately recovers the physics parameter and satisfies the constraint with a small constraint force ($\lambda$). The strong form and energy loss lead to different interpolations when driven only by constraint forces (right).
• Figure 5: Comparing the performance of the strong and weak formulations with the explicit constraint force method in the case that the true model can be exactly recovered. By not working with the strong form residual directly, the weak form incurs larger errors in the recovered constraint force than the strong form.