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Efficient Deconvolution in Populational Inverse Problems

Arnaud Vadeboncoeur, Mark Girolami, Andrew M. Stuart

TL;DR

This work develops a framework for distributional inversion across populations of physical systems where the observational noise is unknown. It jointly learns a parameter distribution for PDEs and the noise distribution by minimizing a Wasserstein-based discrepancy between the data and model distributions, while employing a robust cut-gradient flow and an active-learning surrogate to handle expensive, potentially non-differentiable PDE solvers. The methodology is validated on porous media flow, elastodynamics, and time-averaged chaotic dynamics (Lorenz-96), demonstrating accurate recovery of noise characteristics and model parameters under both uncorrelated and correlated noise and across varied data regimes. The approach promises scalable, data-efficient inference for complex, population-based inverse problems in physics and climate modelling, with practical implications for uncertainty quantification and model calibration in engineering and geoscience contexts.

Abstract

This work is focussed on the inversion task of inferring the distribution over parameters of interest leading to multiple sets of observations. The potential to solve such distributional inversion problems is driven by increasing availability of data, but a major roadblock is blind deconvolution, arising when the observational noise distribution is unknown. However, when data originates from collections of physical systems, a population, it is possible to leverage this information to perform deconvolution. To this end, we propose a methodology leveraging large data sets of observations, collected from different instantiations of the same physical processes, to simultaneously deconvolve the data corrupting noise distribution, and to identify the distribution over model parameters defining the physical processes. A parameter-dependent mathematical model of the physical process is employed. A loss function characterizing the match between the observed data and the output of the mathematical model is defined; it is minimized as a function of the both the parameter inputs to the model of the physics and the parameterized observational noise. This coupled problem is addressed with a modified gradient descent algorithm that leverages specific structure in the noise model. Furthermore, a new active learning scheme is proposed, based on adaptive empirical measures, to train a surrogate model to be accurate in parameter regions of interest; this approach accelerates computation and enables automatic differentiation of black-box, potentially nondifferentiable, code computing parameter-to-solution maps. The proposed methodology is demonstrated on porous medium flow, damped elastodynamics, and simplified models of atmospheric dynamics.

Efficient Deconvolution in Populational Inverse Problems

TL;DR

This work develops a framework for distributional inversion across populations of physical systems where the observational noise is unknown. It jointly learns a parameter distribution for PDEs and the noise distribution by minimizing a Wasserstein-based discrepancy between the data and model distributions, while employing a robust cut-gradient flow and an active-learning surrogate to handle expensive, potentially non-differentiable PDE solvers. The methodology is validated on porous media flow, elastodynamics, and time-averaged chaotic dynamics (Lorenz-96), demonstrating accurate recovery of noise characteristics and model parameters under both uncorrelated and correlated noise and across varied data regimes. The approach promises scalable, data-efficient inference for complex, population-based inverse problems in physics and climate modelling, with practical implications for uncertainty quantification and model calibration in engineering and geoscience contexts.

Abstract

This work is focussed on the inversion task of inferring the distribution over parameters of interest leading to multiple sets of observations. The potential to solve such distributional inversion problems is driven by increasing availability of data, but a major roadblock is blind deconvolution, arising when the observational noise distribution is unknown. However, when data originates from collections of physical systems, a population, it is possible to leverage this information to perform deconvolution. To this end, we propose a methodology leveraging large data sets of observations, collected from different instantiations of the same physical processes, to simultaneously deconvolve the data corrupting noise distribution, and to identify the distribution over model parameters defining the physical processes. A parameter-dependent mathematical model of the physical process is employed. A loss function characterizing the match between the observed data and the output of the mathematical model is defined; it is minimized as a function of the both the parameter inputs to the model of the physics and the parameterized observational noise. This coupled problem is addressed with a modified gradient descent algorithm that leverages specific structure in the noise model. Furthermore, a new active learning scheme is proposed, based on adaptive empirical measures, to train a surrogate model to be accurate in parameter regions of interest; this approach accelerates computation and enables automatic differentiation of black-box, potentially nondifferentiable, code computing parameter-to-solution maps. The proposed methodology is demonstrated on porous medium flow, damped elastodynamics, and simplified models of atmospheric dynamics.

Paper Structure

This paper contains 34 sections, 2 theorems, 45 equations, 25 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions and Outline
  3. Related Works
  4. Deconvolution
  5. Distributional Inversion
  6. Surrogate Modelling
  7. Background
  8. Setup
  9. Distributional Generative Model
  10. Data Distribution
  11. Statistical Divergences
  12. Distributional Inverse Problem
  13. Notation
  14. Methodology
  15. Loss Function
  16. ...and 19 more sections

Key Result

Theorem 3.2

In setting (S) and under Assumption assum:1, the vector field defining both (A1) and (A2) is zero at the global minimizer $(\alpha^\dagger, \Gamma^\dagger)$ of ${\mathsf{J}}_0(\alpha, \Gamma).$

Figures (25)

  • Figure 1: Five solutions $u$ to the porous medium flow problem (dashed lines) and corresponding observation vectors $y$ with 50 observation locations (scatter points).
  • Figure 2: Loss functions \ref{['eq:O1a']} (cut-gradient) with $\gamma'$ fixed at 0.08, and \ref{['eq:O2a']} (standard-gradient). We average the loss function over $100$ sets of 100 projection angles $\theta\sim\mathrm U(\mathbb{S}^{d_y-1})$.
  • Figure 3: Scaled diagonal noise estimation for the porous medium flow problem obtained though stochastic optimisation over $(\alpha, \gamma)$ averaged over 50 runs using \ref{['eq:A1']} (cut-gradient) and \ref{['eq:A2']} (standard-gradient).
  • Figure 4: Five solutions of the porous medium flow problem with Wittle-Matérn process noise with $\beta^\dagger$ parameters $\ell^\dagger=0.25$, $\gamma^\dagger=0.1$, $\upsilon^\dagger=0.5$ and $\alpha$ parameters $m=0.5$, $\sigma=0.5$. (a) shows 5 solution along with observations with spatially correlated corruptions, (b) the data generating noise covariance matrix $\Gamma(\beta^\dagger)$.
  • Figure 5: Whittle-Matérn noise estimation for the porous medium flow problem obtained though stochastic optimisation over $(\alpha, \beta)$ averaged over 100 runs with $\beta=( \gamma^\dagger, \ell, \upsilon^\dagger)$ ($\gamma^\dagger, \upsilon^\dagger$ fixed) using \ref{['eq:A1']} (cut-gradient) and \ref{['eq:A2']} (standard-gradient) for varying $\ell^\dagger$ and $N$.
  • ...and 20 more figures

Theorems & Definitions (7)

  • Theorem 3.2
  • proof
  • Example 3.3
  • Remark 3.4
  • Lemma 6.2
  • proof
  • Remark 6.3