Stochastic Simulation with Geometric Deep Generative Models

Simulation is a key tool in modern science and engineering for studying complex systems and advancing technology. While traditional computational methods such as numerical solvers of differential equations can be highly accurate, they often rely on simplifying assumptions, for example linearity, due to computational constraints or the lack of a complete mathematical model. Recently, neural simulators have emerged as a promising alternative approach. Instead of numerically solving mathematical equations, neural simulators learn to mimic systems directly from data, enabling accuracy improvements and scalable exploration of large parameter spaces. However, current neural simulators face two critical limitations. First, they are typically deterministic, predicting a single outcome for any given input, while many real-life systems are stochastic and require a probabilistic characterization. Second, their black box nature hinders the incorporation of domain knowledge and limits their ability to provide physically meaningful insights beyond simple predictions. This thesis addresses these limitations by developing generative neural simulators that can capture stochastic behavior while integrating with existing scientific knowledge. Our contributions are twofold. First, we explore how domain knowledge can be embedded into generative neural simulators. We introduce two probabilistic models, one autoregressive and one based on Markov Chain Monte Carlo, and demonstrate that incorporating symmetries and other physics-based knowledge improves simulation quality. Second, we apply generative neural simulators to two stochastic systems that lack accurate mathematical models but for which real-world measurements are available: cell-biomaterial interactions and crowd dynamics. We find that generative neural simulators not only reproduce the emergent statistics of these systems, but also facilitate the discovery of new insights through surrogate experiments. In conclusion, this thesis reveals the synergy of physics-based modeling and neural simulation: integrating physical knowledge enhances the fidelity of generative neural simulators, while neural simulators can accelerate the discovery of new insights into poorly understood complex systems.