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Differentiation Strategies for Acoustic Inverse Problems: Admittance Estimation and Shape Optimization

Nikolas Borrel-Jensen, Josiah Bjorgaard

TL;DR

This paper tackles acoustic inverse problems by estimating complex boundary admittance from sparse measurements and by optimizing room geometry to damp resonances, using differentiable physics and differentiable geometry. The authors combine differentiable forward modeling using $JAX$-FEM with a two-stage, hybrid geometry optimization that couples randomized finite differences on the boundary with AD-driven interior mesh updates. Key results include recovering the complex boundary admittance to three-digit accuracy and achieving a $48.1\%$ reduction in resonant energy with a ~30-fold reduction in PDE solves compared to full finite-difference approaches. The work demonstrates how to select differentiation strategies that leverage problem structure for rapid prototyping of physics-based inverse problems.

Abstract

We demonstrate a practical differentiable programming approach for acoustic inverse problems through two applications: admittance estimation and shape optimization for resonance damping. First, we show that JAX-FEM's automatic differentiation (AD) enables direct gradient-based estimation of complex boundary admittance from sparse pressure measurements, achieving 3-digit precision without requiring manual derivation of adjoint equations. Second, we apply randomized finite differences to acoustic shape optimization, combining JAX-FEM for forward simulation with PyTorch3D for mesh manipulation through AD. By separating physics-driven boundary optimization from geometry-driven interior mesh adaptation, we achieve 48.1% energy reduction at target frequencies with 30-fold fewer FEM solutions compared to standard finite difference on the full mesh. This work showcases how modern differentiable software stacks enable rapid prototyping of optimization workflows for physics-based inverse problems, with automatic differentiation for parameter estimation and a combination of finite differences and AD for geometric design.

Differentiation Strategies for Acoustic Inverse Problems: Admittance Estimation and Shape Optimization

TL;DR

This paper tackles acoustic inverse problems by estimating complex boundary admittance from sparse measurements and by optimizing room geometry to damp resonances, using differentiable physics and differentiable geometry. The authors combine differentiable forward modeling using -FEM with a two-stage, hybrid geometry optimization that couples randomized finite differences on the boundary with AD-driven interior mesh updates. Key results include recovering the complex boundary admittance to three-digit accuracy and achieving a reduction in resonant energy with a ~30-fold reduction in PDE solves compared to full finite-difference approaches. The work demonstrates how to select differentiation strategies that leverage problem structure for rapid prototyping of physics-based inverse problems.

Abstract

We demonstrate a practical differentiable programming approach for acoustic inverse problems through two applications: admittance estimation and shape optimization for resonance damping. First, we show that JAX-FEM's automatic differentiation (AD) enables direct gradient-based estimation of complex boundary admittance from sparse pressure measurements, achieving 3-digit precision without requiring manual derivation of adjoint equations. Second, we apply randomized finite differences to acoustic shape optimization, combining JAX-FEM for forward simulation with PyTorch3D for mesh manipulation through AD. By separating physics-driven boundary optimization from geometry-driven interior mesh adaptation, we achieve 48.1% energy reduction at target frequencies with 30-fold fewer FEM solutions compared to standard finite difference on the full mesh. This work showcases how modern differentiable software stacks enable rapid prototyping of optimization workflows for physics-based inverse problems, with automatic differentiation for parameter estimation and a combination of finite differences and AD for geometric design.

Paper Structure

This paper contains 13 sections, 6 equations, 2 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Theory
  3. Acoustic Forward Model
  4. Inverse Problem Formulation
  5. Gradient-Based Optimization with Autodifferentiation
  6. Geometry optimization
  7. Results
  8. Admittance estimation
  9. Room geometry optimization
  10. Conclusion
  11. Technical Appendices and Supplementary Material
  12. Finite Element Discretization
  13. Two-Stage Randomized Gradient Algorithm for Shape Optimization

Figures (2)

  • Figure 1: Convergence of the normalized admittance estimation. The initial guess was $\beta_{\text{est}} = 3.0 + 0.05$ and the method estimated the real admittance $\beta_{\text{true}}$ within 3 digits.
  • Figure 2: Results of room geometry optimization w.r.t. acoustic energy over 100 steps, showing total acoustic energy (scaled by $10^{-8}$) and mesh loss (scaled by $10^2$). Initial mesh and $|p|$ field and mesh geometry are shown along with mesh and $|p|$ at step 100.