Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Differential-Equation Constrained Optimization With Stochasticity

Qin Li, Li Wang, Yunan Yang

TL;DR

This paper conceptualizes the PDE solver as a push-forward map that pushes the parameter distribution to the generated data distribution, and forms a gradient-flow equation to seek the ground-truth parameter probability distribution.

Abstract

Most inverse problems from physical sciences are formulated as PDE-constrained optimization problems. This involves identifying unknown parameters in equations by optimizing the model to generate PDE solutions that closely match measured data. The formulation is powerful and widely used in many sciences and engineering fields. However, one crucial assumption is that the unknown parameter must be deterministic. In reality, however, many problems are stochastic in nature, and the unknown parameter is random. The challenge then becomes recovering the full distribution of this unknown random parameter. It is a much more complex task. In this paper, we examine this problem in a general setting. In particular, we conceptualize the PDE solver as a push-forward map that pushes the parameter distribution to the generated data distribution. This way, the SDE-constrained optimization translates to minimizing the distance between the generated distribution and the measurement distribution. We then formulate a gradient-flow equation to seek the ground-truth parameter probability distribution. This opens up a new paradigm for extending many techniques in PDE-constrained optimization to that for systems with stochasticity.

Differential-Equation Constrained Optimization With Stochasticity

TL;DR

This paper conceptualizes the PDE solver as a push-forward map that pushes the parameter distribution to the generated data distribution, and forms a gradient-flow equation to seek the ground-truth parameter probability distribution.

Abstract

Most inverse problems from physical sciences are formulated as PDE-constrained optimization problems. This involves identifying unknown parameters in equations by optimizing the model to generate PDE solutions that closely match measured data. The formulation is powerful and widely used in many sciences and engineering fields. However, one crucial assumption is that the unknown parameter must be deterministic. In reality, however, many problems are stochastic in nature, and the unknown parameter is random. The challenge then becomes recovering the full distribution of this unknown random parameter. It is a much more complex task. In this paper, we examine this problem in a general setting. In particular, we conceptualize the PDE solver as a push-forward map that pushes the parameter distribution to the generated data distribution. This way, the SDE-constrained optimization translates to minimizing the distance between the generated distribution and the measurement distribution. We then formulate a gradient-flow equation to seek the ground-truth parameter probability distribution. This opens up a new paradigm for extending many techniques in PDE-constrained optimization to that for systems with stochasticity.
Paper Structure (26 sections, 7 theorems, 115 equations, 7 figures, 1 algorithm)

This paper contains 26 sections, 7 theorems, 115 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Comparisons with Other Subjects
  3. Bayesian Inverse Problem
  4. Density Estimation
  5. Gradient Flow Formulation and Particle Methods
  6. The Wasserstein Gradient Flow Strategy
  7. Other Possible Metrics
  8. Particle Method
  9. Adjoint Solver Simplification
  10. Discussion on the Particle Method
  11. Well-posedness Result for Linear Push-Forward Operators
  12. Under-Determined Scenario
  13. Deterministic Case
  14. Stochastic Case
  15. Over-Determined Scenario
  16. ...and 11 more sections

Key Result

Proposition 1

\newlabelprop:under_determine0 The equilibrium solution to eqn:theta_ODE_determ, denoted by $u_\text{f}$, given the initial iterate $u_\text{i}$, can be written as Moreover, $y^\mathrm{ex}_\text{f} = \mathsf{A}^\mathrm{ex} u_\text{f}$ as defined in eqn:def_A_ex satisfies:

Figures (7)

  • Figure 1: Parameter and data distributions in the fully-determined case with the map $T(x) = \mathsf{A}x$ where $\mathsf{A} = \text{diag}([2,0.75])$.
  • Figure 2: Under-determined case under two initial distributions, $u^1$ and $u^2$, where the map $\mathcal{G}(u) = \mathsf{A}u$, $\mathsf{A} = [2, 0.75]$. The reference data distribution is computed using the true parameter distribution.
  • Figure 3: Over-determined case with the map $T(x) = \mathsf{A}x$ where $\mathsf{A} = [2, 1]^\top$. Although the final recovered data distribution does not fit the reference entirely, their marginal distributions on $y_A$ match very well, as proved in \ref{['thm:over_determin_stoc']}.
  • Figure 4: Numerical inversion based on the 1D diffusion equation \ref{['eq:1D-eit']} with setting (1).
  • Figure 5: Numerical inversion based on the 1D diffusion equation \ref{['eq:1D-eit']} with setting (2).
  • ...and 2 more figures

Theorems & Definitions (16)

  • Definition 1: Kantorovich formulation
  • Definition 2: The Hellinger distance
  • Proposition 1
  • Theorem 2
  • Corollary 3
  • Lemma 4
  • Proof 1
  • Lemma 5
  • Proof 2
  • Remark 1
  • ...and 6 more