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Publication# PDE-Aware Deep Learning for Inverse Problems in Cardiac Electrophysiology

Résumé

In this work, we present a PDE-aware deep learning model for the numerical solution to the inverse problem of electrocardiography. The model both leverages data availability and exploits the knowledge of a physically based mathematical model, expressed by means of partial differential equations (PDEs), to carry out the task at hand. The goal is to estimate the epicardial potential field from measurements of the electric potential at a discrete set of points on the body surface. The employment of deep learning techniques in this context is made difficult by the low amount of clinical data at disposal, as measuring cardiac potentials requires invasive procedures. Suitably exploiting the underlying physically based mathematical model allowed circumventing the data availability issue and led to the development of fast-training and low-complexity models. Physical awareness has been pursued by means of two elements: the projection of the epicardial potential onto a space-time reduced subspace, spanned by the numerical solutions of the governing PDEs, and the inclusion of a tensorial reduced basis solver of the forward problem in the network architecture. Numerical tests have been conducted only on synthetic data, obtained via a full order model approximation of the problem at hand, and two variants of the model have been addressed. Both proved to be accurate, up to an average $\ell^1$-norm relative error on epicardial activation maps of 3.5%, and both could be trained in \approx$$15 min. Nevertheless, some improvements, mostly concerning data generation, are necessary in order to bridge the gap with clinical applications.

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Numerical mathematics is the branch of mathematics that proposes, develops, analyzes and applies methods from scientific computing to several fields including analysis, linear algebra, geometry, approximation theory, functional equations, optimization and differential equations. Other disciplines, such as physics, the natural and biological sciences, engineering, and economics and the financial sciences frequently give rise to problems that need scientific computing for their solutions. As such, numerical mathematics is the crossroad of several disciplines of great relevance in modern applied sciences, and can become a crucial tool for their qualitative and quantitative analysis. One of the purposes of this book is to provide the mathematical foundations of numerical methods, to analyze their basic theoretical properties (stability, accuracy, computational complexity) and demonstrate their performances on examples and counterexamples which outline their pros and cons. This is done using the MATLAB software environment which is user-friendly and widely adopted. Within any specific class of problems, the most appropriate scientific computing algorithms are reviewed, their theoretical analyses are carried out and the expected results are verified on a MATLAB computer implementation. Every chapter is supplied with examples, exercises and applications of the discussed theory to the solution of real-life problems. This book is addressed to senior undergraduate and graduate students with particular focus on degree courses in Engineering, Mathematics, Physics and Computer Sciences. The attention which is paid to the applications and the related development of software makes it valuable also for researchers and users of scientific computing in a large variety of professional fields. In this second edition, the readability of pictures, tables and program headings have been improved. Several changes in the chapters on iterative methods and on polynomial approximation have also been added.

In this thesis, we propose model order reduction techniques for high-dimensional PDEs that preserve structures of the original problems and develop a closure modeling framework leveraging the Mori-Zwanzig formalism and recurrent neural networks. Since high-fidelity approximations of PDEs often result in a large number of degrees of freedom, the need for iterative evaluations for numerical optimizations and rapid feedback is computationally challenging.The first part of this thesis is devoted to conserving the high-dimensional equation's invariants, symmetries, and structures during the reduction process. Traditional reduction techniques are not guaranteed to yield stable reduced systems, even if the target problem is stable. In the context of fluid flows, the skew-symmetric structure of the problem entails the preservation of the kinetic energy of the system. By preserving the same structure at the level of the reduced model, we obtain enhanced stability, and accuracy and the reduced model acquires physical significance by preserving a surrogate of the energy of the original problem. Next, we focus on Hamiltonian systems, which, being driven by symmetry, are a source of great interest in the reduction community. It is well known that the breaking of these symmetries in the reduced model is accompanied by a blowup of the system energy and flow volume. In this thesis, geometric reduced models for Hamiltonian systems are further developed and combined with the dynamically orthogonal methods, addressing the poor reducibility in time of advection-dominated problems. The reduced solution is expressed as a linear combination of a finite number of modes and coincides with the symplectic projection of the high-fidelity Hamiltonian problem onto the tangent space of the approximating manifold. An error surrogate is used to monitor the approximation ability of the reduced model and make a change in the rank of the approximating system if necessary. The method is further developed through a combination of DEIM and DMD to reduce non-polynomial nonlinearities while preserving the symplectic structure of the problem and applied to the Vlasov-Poisson system.In the second part of the thesis, we consider several data-driven methods to address the poor accuracy in the under-resolved regime for Galerkin reduced models via a closure term. The closure term is developed systematically from the Mori-Zwanzig formalism by introducing projection operators on the spaces of resolved and unresolved scales, thus resulting in an additional memory integral term. The interaction between different scales turns out to be nonlocal in time and dominated by a high-dimensional orthogonal dynamics equation, which cannot be solved precisely and efficiently. Several classical methods in the field of statistical mechanics are used to approximate the memory term, exploiting the finiteness of the memory kernel support. We conclude this thesis by showing through numerical experiments how long short-term memory networks, i.e., machine learning structures characterized by feedback connections, represent a valid tool for approximating the additional memory term.

In this thesis we explore uncertainty quantification of forward and inverse problems involving differential equations. Differential equations are widely employed for modeling natural and social phenomena, with applications in engineering, chemistry, meteorology, and economics. Mathematical models of complex systems in these fields require numerical methods, which introduce uncertainties in the outcome. Moreover, there has recently been a steep rise in the availability of data, which also come with an uncertainty. Therefore, blending mathematical models, data, and their respective uncertainties is nowadays of the utmost importance. The first part of this thesis is dedicated to two novel methods for multiscale inverse problems. We first consider an elliptic partial differential equation (PDE), with a diffusion tensor oscillating at a small scale. Given noisy observations of the solution, we consider the problem of inferring a slow-scale parametrization of the multiscale tensor. For this purpose, we combine numerical homogenization, which yields a single-scale surrogate of the full model, and the ensemble Kalman filter. The scheme we propose is accurate in the homogenized limit, and outperforms existing methods in terms of computational cost. We then study the error due to the mismatch between the full and the homogenized models, and show how to combine statistical techniques for model misspecification and our scheme. We then move to multiscale diffusion processes, and consider the problem of inferring effective dynamics from multiscale observations. A homogenized single-scale equation reproducing the full model exists also in this case. The effective model, though, is the subject of the inference procedure, and not only a computational tool. The resulting issue of model misspecification is usually bypassed by subsampling at an appropriate rate, which is non-trivial to choose, and which may give misleading results. We avoid subsampling by designing a novel technique based on filtered data, and show how to modify classical estimators and obtain an effective equation consistently with homogenization. Our technique is robust and can be employed as a black-box tool for inferring effective surrogates of complex stochastic models.In the second part we present two novel schemes belonging to the field of probabilistic numerics, whose purpose is to provide a statistical description of the uncertainty due to numerical discretization. We first consider ordinary differential equations (ODEs), and introduce a probabilistic integrator based on random time steps and Runge-Kutta methods (RTS-RK). Tuning the distribution of the time steps, we generate a probability measure on the solution which allows for a consistent uncertainty quantification of numerical errors. Unlike previous probabilistic methods in literature, our scheme inherits the geometric properties of the underlying deterministic integrators. In particular, we show long-time energy conservation when the RTS-RK is applied to Hamiltonian ODEs. We employ the idea of randomizing the discretization to propose a random mesh finite element method (RM-FEM) for elliptic PDEs. We prove that the measure induced by the RM-FEM on the solution can be employed to derive a posteriori error estimators. Hence, the RM-FEM provides a consistent statistical characterization of numerical errors. For both our novel schemes, we demonstrate the usefulness of the probabilistic approach in Bayesian inverse problems.