Abstract

Scientific simulations are central to understanding complex physical systems, informing engineering design, risk assessment, and scientific discovery. However, traditional numerical solvers scale poorly as we increase resolution, consider high-dimensional domains, or attempt to capture multi-scale physics, leading to prohibitive computational cost. Recent data-driven approaches offer a different philosophy: rather than re-solving the governing equations from scratch, we aim to learn the underlying physics directly from data. Physics-informed neural networks (PINNs) were an early step in this direction, embedding soft differential equation constraints into the loss function. However, they face challenges with optimisation, stiffness, and scalability to large domains and long time horizons, and they must be retrained for each new boundary or initial condition. In this talk, I will discuss neural operators as a promising alternative. Neural operators learn mappings between infinite-dimensional function spaces, enabling mesh-free inference, efficient resolution refinement, and amortised solution of many-query simulation tasks. These models have shown strong performance on high-dimensional, non-linear, and multi-scale phenomena, from weather forecasting to single-cell electrophysiology and photonics, and I will highlight both current capabilities and open challenges.