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Goal-Oriented Real-Time Bayesian Inference for Linear Autonomous Dynamical Systems With Application to Digital Twins for Tsunami Early Warning

Stefan Henneking, Sreeram Venkat, Omar Ghattas

TL;DR

This paper develops a goal-oriented Bayesian digital twin framework for real-time inversion of large-scale linear autonomous PDEs, targeting tsunami early warning from subduction zones. By exploiting time-shift invariance, it decomposes the computation into offline precomputations and an online phase that avoids PDE solves, using FFT-based, multi-GPU matvecs for the p2o and p2q maps and their adjoints. The approach rewrites the Hessian inverse via the Sherman–Morrison–Woodbury formula to shift the inversion to data space, enabling exact posterior contractions and efficient QoI uncertainty propagation through a data-to-QoI map, with Phase 3 computing QoI covariances and Phase 4 enabling real-time parameter inference and QoI prediction. Demonstrations on a representative Cascadia-scale problem show end-to-end inference and prediction for over $10^8$ parameters and hundreds of QoIs in fractions of a second, illustrating a viable path to real-time digital twins for tsunami early warning. The framework is general and extensible to other autonomous PDE-driven systems beyond tsunami scenarios, promising significant impact on rapid, uncertainty-quantified decision making in critical sensing applications.

Abstract

We present a goal-oriented framework for constructing digital twins with the following properties: (1) they employ discretizations of high-fidelity PDE models governed by autonomous dynamical systems, leading to large-scale forward problems; (2) they solve a linear inverse problem to assimilate observational data to infer uncertain model components followed by a forward prediction of the evolving dynamics; and (3) the entire end-to-end, data-to-inference-to-prediction computation is carried out without approximation and in real time through a Bayesian framework that rigorously accounts for uncertainties. Several challenges must be overcome to realize this framework, including the large scale of the forward problem, the high dimensionality of the parameter space, and for a class of problems including those we target, the slow decay of the singular values of the parameter-to-observable map. Here we introduce a methodology to overcome these challenges by exploiting the autonomous structure of the forward model to decompose the solution of the inverse problem into an offline phase in which the PDE model is solved a limited number of times, and an online phase that computes the parameter inference and prediction of quantities of interest in real time, given observational data. Our goal is to apply this framework to construct digital twins for subduction zones to provide early warning for tsunamis. To this end, we show how our methodology can be used to employ seafloor pressure observations, along with the coupled acoustic-gravity wave equations, to infer the earthquake-induced seafloor motion (discretized with 10^9 parameters) and forecast the tsunami propagation. We present results of an end-to-end inference, prediction, and uncertainty quantification for a representative test problem with 10^8 inversion parameters for which goal-oriented Bayesian inference is accomplished in real time.

Goal-Oriented Real-Time Bayesian Inference for Linear Autonomous Dynamical Systems With Application to Digital Twins for Tsunami Early Warning

TL;DR

This paper develops a goal-oriented Bayesian digital twin framework for real-time inversion of large-scale linear autonomous PDEs, targeting tsunami early warning from subduction zones. By exploiting time-shift invariance, it decomposes the computation into offline precomputations and an online phase that avoids PDE solves, using FFT-based, multi-GPU matvecs for the p2o and p2q maps and their adjoints. The approach rewrites the Hessian inverse via the Sherman–Morrison–Woodbury formula to shift the inversion to data space, enabling exact posterior contractions and efficient QoI uncertainty propagation through a data-to-QoI map, with Phase 3 computing QoI covariances and Phase 4 enabling real-time parameter inference and QoI prediction. Demonstrations on a representative Cascadia-scale problem show end-to-end inference and prediction for over parameters and hundreds of QoIs in fractions of a second, illustrating a viable path to real-time digital twins for tsunami early warning. The framework is general and extensible to other autonomous PDE-driven systems beyond tsunami scenarios, promising significant impact on rapid, uncertainty-quantified decision making in critical sensing applications.

Abstract

We present a goal-oriented framework for constructing digital twins with the following properties: (1) they employ discretizations of high-fidelity PDE models governed by autonomous dynamical systems, leading to large-scale forward problems; (2) they solve a linear inverse problem to assimilate observational data to infer uncertain model components followed by a forward prediction of the evolving dynamics; and (3) the entire end-to-end, data-to-inference-to-prediction computation is carried out without approximation and in real time through a Bayesian framework that rigorously accounts for uncertainties. Several challenges must be overcome to realize this framework, including the large scale of the forward problem, the high dimensionality of the parameter space, and for a class of problems including those we target, the slow decay of the singular values of the parameter-to-observable map. Here we introduce a methodology to overcome these challenges by exploiting the autonomous structure of the forward model to decompose the solution of the inverse problem into an offline phase in which the PDE model is solved a limited number of times, and an online phase that computes the parameter inference and prediction of quantities of interest in real time, given observational data. Our goal is to apply this framework to construct digital twins for subduction zones to provide early warning for tsunamis. To this end, we show how our methodology can be used to employ seafloor pressure observations, along with the coupled acoustic-gravity wave equations, to infer the earthquake-induced seafloor motion (discretized with 10^9 parameters) and forecast the tsunami propagation. We present results of an end-to-end inference, prediction, and uncertainty quantification for a representative test problem with 10^8 inversion parameters for which goal-oriented Bayesian inference is accomplished in real time.
Paper Structure (28 sections, 38 equations, 5 figures, 3 tables)

This paper contains 28 sections, 38 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Large-scale inverse problems
  4. Contributions
  5. Background
  6. Notation
  7. Linear time-invariant dynamical system
  8. Discrete LTI system
  9. Bayesian inverse problem
  10. Goal-oriented Bayesian inference
  11. Tsunami Inverse Model
  12. Acoustic--gravity wave equations
  13. Adjoint equations
  14. Hessian action
  15. Discretization of the CSZ tsunami inverse problem
  16. ...and 13 more sections

Figures (5)

  • Figure 1: 3D spatial domain of a test configuration with the locations of 49 seafloor pressure sensors and 16 predicted QoIs (tsunami wave heights).
  • Figure 2: Eigenvalues of $\mathbf{F} \mathbf{F}^*$ for the test configuration. The slow eigenvalue decay is indicative of hyperbolic PDE operators which do not readily admit compact representations.
  • Figure 3: Total seafloor displacement (time-integrated parameter field). Left: True field; Middle: Inferred mean from synthetic observations with 6% relative additive noise; Right: Error between true and inferred fields.
  • Figure 4: Surface gravity wave. Left: True field; Middle: Reconstructed field based on inferred mean of seafloor motion (from synthetic observations with 6% relative additive noise); Right: Error between true and reconstructed surface gravity wave based on inferred mean of seafloor motion.
  • Figure 5: True and predicted QoIs (tsunami wave heights at specified locations) with uncertainties depicted in the form of 95% credible intervals (CIs). Left: QoI locations #1--3; Right: QoI locations #5--6 (cf. Figure \ref{['fig:domain']}).

Theorems & Definitions (1)

  • Remark