Abstract
High-fidelity numerical simulations are essential for understanding extreme physical phenomena. Yet, at a large scale, their practical use is often constrained by the prohibitive computational cost of resolving highly nonlinear spatiotemporal dynamics. Physics-aware deep learning (PADL) has emerged as a promising surrogate, offering the potential to accelerate such simulations by orders of magnitude. To date, however, most PADL approaches neglect the adaptive discretization strategies that are foundational to robust numerical analysis. Instead, they rely on ``off-the-shelf'' architectures adapted from computer vision, inheriting rigid assumptions such as fixed Eulerian (pixel-based) grids and constant time-steps (frame rates). This rigidity stands in direct contrast to numerical literature, where adaptive spatial and temporal resolutions play an essential role for resolving sharp gradients, fast transients, and discontinuities. As a result, PADL models often trade physical fidelity and interpretability for efficiency, limiting their ability to generalize to the stiff, nonlinear regimes encountered in real-world applications.
The central contribution of this dissertation is the first attempt to integrate adaptive discretization principles into PADL, yielding physics-aware neural architectures that dynamically adjust their spatial and temporal resolutions to the underlying physics. We develop this work by advancing physics-aware recurrent convolutional neural networks (PARC) from a rigid Eulerian formulation to hybrid Lagrangian-Eulerian (HLE) adaptive refinement and unstructured spatial discretization. In particular, we introduce three complementary mechanisms that bridge the gap between PADL and the numerical literature. First, we incorporate deformable convolutions to track advecting features, effectively embedding a Lagrangian reference frame within an Eulerian architecture (D-PARC). Second, we develop a variable time-integration strategy that enables stable learning across irregular time steps to resolve transient dynamics. Third, we generalize physics-aware convolutions to unstructured domains within a graph neural network message-passing scheme (G-PARC).
From a practical perspective, the incorporation of adaptivity yields interpretable surrogate solvers that more closely reflect the structure of their numerical counterparts. By mitigating the numerical error inherent in static discretizations and ``off-the-shelf'' architectures, while respecting irregular geometries and boundary conditions, our approach enables robust acceleration for challenging physical systems, including compressible, incompressible, and reactive flows, fluvial hydrology, planar shock wave dynamics, and structural mechanics.
