Extending the explicit constraint force method to inverse problems

TL;DR

This work extends the Explicit Constraint Force Method (ECFM) from solution reconstruction to deterministic inverse problems, comparing it against standard PDE-constrained formulations. It demonstrates three numerical scenarios—dynamic, noisy, and stochastic—to assess the viability and behavior of constraint forces as a data-consistency mechanism. For noisy data, it introduces inequality constraints that align discrepancy statistics with measurement noise, and for stochastic models it employs polynomial chaos expansion with a pseudo-likelihood objective. The results show that ECFM can recover model parameters with competitive accuracy, reveal missing physics via constraint forces, and extend naturally to parametric boundaries and domain geometries, indicating its potential as a principled inverse-analysis tool.

Abstract

Recently, the explicit constraint force method (ECFM) was introduced as a principled approach to solution reconstruction in the presence of missing physics. In solution reconstruction, parameters of a physical model are estimated from sparse measurement data as a means to obtain the full solution field. In contrast, inverse problems target the missing parameters and estimate the solution along the way. Noting the similarity of the mathematical formulations of these two tasks, we investigate the use of ECFM to solve inverse problems. First, we compare the ECFM formulation of the inverse problem to a standard approach using two numerical examples. The first example provides an extension of ECFM to dynamic problems, and the second offers a novel approach to treat noisy measurement data. Next, we introduce a method to solve inverse problems for which the parameterized model has stochastic components. This approach is based on constraint forces and the polynomial chaos expansion, and is illustrated with another numerical example. Finally, we discuss extensions of ECFM to recover missing boundary conditions and domain geometries from measurement data, which are shown to be special cases of problems treated previously in the literature. The purpose of this work is to extend the mathematical framework of ECFM to novel applications and to gauge the method's viability as an alternative strategy for inverse analysis.

Figures (10)

• Figure 1: Problem setup for extracting the viscosity and source term magnitude from flow data generated by solving Burgers' equation.
• Figure 2: True solution and the time series of the $C=4$ measurements taken, which are used in the inverse problem to recover the viscosity and source term magnitude.
• Figure 3: The standard inverse problem converges to a zero-error solution because the parameterized model is consistent with the measurement data (left). The loss surface is convex in the vicinity of the true solution, indicating an identifiable inverse problem (right).
• Figure 4: The constraint force formulation of the inverse problem converges to model parameters with zero associated constraint force because the parameterized model is consistent with the measurement data (left). The loss surface representing the constraint force magnitude is also convex in the vicinity of the true solution, indicating an identifiable inverse problem (right).
• Figure 5: The parameterized model is inconsistent with the measurement data because the true source term cannot be recovered by a finite sine series (left). The true solution is generated by approximating a solution to Eq. \ref{['kpp']} with the discontinuous source term (right). Measurement data is taken at $C=225$ locations with normally distributed noise of a given mean and variance.