Latent Twins for Bathymetry Inversion Open Access

Abstract

Recovering spatially varying coefficients, such as bathymetry, from sparse state observations is a notoriously ill-posed inverse problem in geophysical fluid dynamics. While Variational Data Assimilation offers a rigorous solution, it relies on computationally expensive adjoint loops that are prohibitive for real-time applications. Conversely, standard deep learning surrogates provide speed but frequently fail to capture the causal structure of the governing Partial Differential Equations, leading to physically inconsistent solutions. This thesis proposes the Latent Twin framework to unify inversion and forecasting through joint state-parameter estimation, demonstrating its application to the Shallow Water Equations. We construct a dual-manifold architecture comprising a dynamic State Twin and a static Parameter Twin, coupled via two learned operators: a latent mapping that performs inversion, and a parameter-conditioned transition operator. This coupling enables a fully joint estimation strategy: the inferred bathymetry serves as the necessary physical coefficients to evolve the latent flow, while the requirement to accurately forecast wave propagation acts as a dynamic consistency constraint. By enforcing this constraint, we implicitly regularize the ill-posed inverse solution without requiring analytical priors. We demonstrate that this architecture recovers bathymetry with high fidelity and enables inversion orders of magnitude faster than iterative solvers, demonstrating robust generalization across diverse bathymetric topologies and roughness.

Table of Contents

1 Introduction 1

1.1 General Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Forward and Inverse . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Practical Joint Treatment . . . . . . . . . . . . . . . . . . . 6

1.4 Fundamental Challenges . . . . . . . . . . . . . . . . . . . .7

1.4.1 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.2 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Data Assimilation Methods . . . . . . . . . . . . . . . . . 11

1.5.1 Variational Data Assimilation (4D-Var) . . . . 14

1.5.2 EnKF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

1.5.3 Differentiable Physics Surrogates . . . . . . . . . .15

1.6 Summary of Methods . . . . . . . . . . . . . . . . . . . . . . . .16

1.7 Thesis Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

2 SWE and Bathymetry 21

2.1 The Shallow Water Equations (SWE) . . . . . . . . . 21

2.1.1 Bathymetry in the SWE . . . . . . . . . . . . . . . . . . . . 22

2.1.2 Additional Considerations of the SWE . . . . . .23

2.2 Model Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25

i

2.3 Joint State-Space Formulation . . . . . . . . . . . . . . . 26

3 Latent Twins 28

3.1 Latent Twins at Large . . . . . . . . . . . . . . . . . . . . . . . .28

3.2 Methodology, Architecture . . . . . . . . . . . . . . . . . . .31

3.3 Connection to DA Methods . . . . . . . . . . . . . . . . . . .34

3.4 Loss Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37

4 Numerical Experiments 40

4.1 Experiment 1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40

4.1.1 Data Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . .41

4.1.2 Architecture and Training . . . . . . . . . . . . . . . . . . 42

4.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2 Experiment 2: Generalization . . . . . . . . . . . . . . . . .49

4.2.1 Data Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.2 Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2.3 Loss Functional and Three-Phase Training . . 52

4.2.4 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 54

5 Conclusion 58

5.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2 Scalability and Data Dependency . . . . . . . . . . . . . . .60

5.3 Limitations and Future Work . . . . . . . . . . . . . . . . . . .61

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Bibliography 64

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Applied Mathematics Engineering, Marine and Ocean
Keyword	Bathymetry Inverse Problem Data Assimilation Latent Twins
Committee Chair / Thesis Advisor	Mattias Chung, Emory University
Committee Members	Marco Tezzele, Emory University Alessandro Veneziani, Emory University

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