Deep learning-based surrogate models for parametrized PDEs: Handling geometric variability through graph neural networks.

Mesh-based simulations play a key role when modeling complex physical systems that, in many disciplines across science and engineering, require the solution to parametrized time-dependent nonlinear partial differential equations (PDEs). In this context, full order models (FOMs), such as those relying on the finite element method, can reach high levels of accuracy, however often yielding intensive simulations to run. For this reason, surrogate models are developed to replace computationally expensive solvers with more efficient ones, which can strike favorable trade-offs between accuracy and efficiency.

Efficient approximation of cardiac mechanics through reduced-order modeling with deep learning-based operator approximation.

Reducing the computational time required by high-fidelity, full-order models (FOMs) for the solution of problems in cardiac mechanics is crucial to allow the translation of patient-specific simulations into clinical practice. Indeed, while FOMs, such as those based on the finite element method, provide valuable information on the cardiac mechanical function, accurate numerical results can be obtained at the price of very fine spatio-temporal discretizations. As a matter of fact, simulating even just a few heartbeats can require up to hours of wall time on high-performance computing architectures.

Reduced Order Modeling of Nonlinear Vibrating Multiphysics Microstructures with Deep Learning-Based Approaches.

Micro-electro-mechanical-systems are complex structures, often involving nonlinearites of geometric and multiphysics nature, that are used as sensors and actuators in countless applications. Starting from full-order representations, we apply deep learning techniques to generate accurate, efficient, and real-time reduced order models to be used for the simulation and optimization of higher-level complex systems. We extensively test the reliability of the proposed procedures on micromirrors, arches, and gyroscopes, as well as displaying intricate dynamical evolutions such as internal resonances.

Approximation bounds for convolutional neural networks in operator learning.

Recently, deep Convolutional Neural Networks (CNNs) have proven to be successful when employed in areas such as reduced order modeling of parametrized PDEs. Despite their accuracy and efficiency, the approaches available in the literature still lack a rigorous justification on their mathematical foundations. Motivated by this fact, in this paper we derive rigorous error bounds for the approximation of nonlinear operators by means of CNN models.

POD-Enhanced Deep Learning-Based Reduced Order Models for the Real-Time Simulation of Cardiac Electrophysiology in the Left Atrium.

The numerical simulation of multiple scenarios easily becomes computationally prohibitive for cardiac electrophysiology (EP) problems if relying on usual high-fidelity, full order models (FOMs). Likewise, the use of traditional reduced order models (ROMs) for parametrized PDEs to speed up the solution of the aforementioned problems can be problematic. This is primarily due to the strong variability characterizing the solution set and to the nonlinear nature of the input-output maps that we intend to reconstruct numerically.

Deep learning-based reduced order models in cardiac electrophysiology.

Predicting the electrical behavior of the heart, from the cellular scale to the tissue level, relies on the numerical approximation of coupled nonlinear dynamical systems. These systems describe the cardiac action potential, that is the polarization/depolarization cycle occurring at every heart beat that models the time evolution of the electrical potential across the cell membrane, as well as a set of ionic variables. Multiple solutions of these systems, corresponding to different model inputs, are required to evaluate outputs of clinical interest, such as activation maps and action potential duration.