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The explicit constraint force method for optimal experimental design

Conor Rowan

TL;DR

This work assesses the viability of designing experiments via the explicit constraint force method (ECFM) for inverse problems governed by PDEs. By recasting OED in a deterministic (Gauss-Newton) framework and then applying ECFM, the authors show that optimal sensor placement tends to concentrate in the system's stiff regions, which is impractical in the presence of noise and finite precision. Across several analytically tractable test cases, the constraint-force-based design often contradicts standard Fisher-information-based recommendations, sometimes placing measurements at domain boundaries where physical measurements are challenging. The provisional conclusion is that ECFM is not a viable approach for OED, though the work clarifies the distinct notions of discrepancy (displacement versus force) and illuminates when constraint-force formulations may be appropriate, such as in other aspects of inverse problems. The findings guide future work toward alternative, robust OED strategies that account for measurement realism and noise characteristics.

Abstract

The explicit constraint force method (ECFM) was recently introduced as a novel formulation of the physics-informed solution reconstruction problem, and was subsequently extended to inverse problems. In both solution reconstruction and inverse problems, model parameters are estimated with the help of measurement data. In practice, experimentalists seek to design experiments such that the acquired data leads to the most robust recovery of the missing parameters in a subsequent inverse problem. While there are well-established techniques for designing experiments with standard approaches to the inverse problem, optimal experimental design (OED) has yet to be explored with the ECFM formulation. In this work, we investigate OED with a constraint force objective. First, we review traditional approaches to OED based on the Fisher information matrix, and propose an analogous formulation based on constraint forces. Next, we reflect on the different interpretations of the objective from standard and constraint force-based inverse problems. We then test our method on several example problems. These examples suggest that an experiment which is optimal in the sense of constraint forces tends to position measurements in the stiffest regions of the system. Because the responses -- and thus the measurements -- are small in these regions, this strategy is impractical in the presence of measurement noise and/or finite measurement precision. As such, our provisional conclusion is that ECFM is not a viable approach to OED.

The explicit constraint force method for optimal experimental design

TL;DR

This work assesses the viability of designing experiments via the explicit constraint force method (ECFM) for inverse problems governed by PDEs. By recasting OED in a deterministic (Gauss-Newton) framework and then applying ECFM, the authors show that optimal sensor placement tends to concentrate in the system's stiff regions, which is impractical in the presence of noise and finite precision. Across several analytically tractable test cases, the constraint-force-based design often contradicts standard Fisher-information-based recommendations, sometimes placing measurements at domain boundaries where physical measurements are challenging. The provisional conclusion is that ECFM is not a viable approach for OED, though the work clarifies the distinct notions of discrepancy (displacement versus force) and illuminates when constraint-force formulations may be appropriate, such as in other aspects of inverse problems. The findings guide future work toward alternative, robust OED strategies that account for measurement realism and noise characteristics.

Abstract

The explicit constraint force method (ECFM) was recently introduced as a novel formulation of the physics-informed solution reconstruction problem, and was subsequently extended to inverse problems. In both solution reconstruction and inverse problems, model parameters are estimated with the help of measurement data. In practice, experimentalists seek to design experiments such that the acquired data leads to the most robust recovery of the missing parameters in a subsequent inverse problem. While there are well-established techniques for designing experiments with standard approaches to the inverse problem, optimal experimental design (OED) has yet to be explored with the ECFM formulation. In this work, we investigate OED with a constraint force objective. First, we review traditional approaches to OED based on the Fisher information matrix, and propose an analogous formulation based on constraint forces. Next, we reflect on the different interpretations of the objective from standard and constraint force-based inverse problems. We then test our method on several example problems. These examples suggest that an experiment which is optimal in the sense of constraint forces tends to position measurements in the stiffest regions of the system. Because the responses -- and thus the measurements -- are small in these regions, this strategy is impractical in the presence of measurement noise and/or finite measurement precision. As such, our provisional conclusion is that ECFM is not a viable approach to OED.
Paper Structure (15 sections, 62 equations, 1 figure, 1 table)

This paper contains 15 sections, 62 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Optimal experimental design
  3. State of the art
  4. A deterministic perspective
  5. Designing experiments for PDE systems
  6. Designing experiments with constraint forces
  7. Two conceptions of the inverse problem
  8. Testing the method
  9. Parameterized boundary condition
  10. Parameterized source term
  11. Parameterized material
  12. Mis-specified model
  13. Discussion
  14. Eigenvalue sensitivity analysis
  15. ECFM sensitivity analysis

Figures (1)

  • Figure 1: In the case of the standard inverse problem, the model parameters $\boldsymbol \epsilon$ are chosen to minimize $\sum_i\Delta u_i^2(\boldsymbol \epsilon)$, which is the displacement discrepancy. With ECFM, the model parameters minimize the magnitude of the constraint force, given by $\sum_i \lambda_i^2(\boldsymbol \epsilon)$.