Advances in physics-informed deep learning

Accurate models of physical systems play a fundamental role in numerous scientific and industrial fields. Traditional models, grounded in physical laws and expressed through partial differential equations (PDEs), are powerful yet limited. These traditional techniques often exhibit slow inference, rely on oversimplified assumptions, and tend to overlook or underutilize the data gathered from the modeled system. While data-driven approaches like neural networks can overcome these limitations, the data collection is typically too scarce to support purely data-driven methods. This thesis addresses the challenge of modeling physical systems with data-driven techniques when the data from the system is limited and scarce. This research work advances modeling methodologies by embedding prior knowledge of the governing physics into various stages of the modeling procedure. First, the thesis proposes using diffusion-based generative models as probabilistic surrogates to recover unobserved states of physical systems, utilizing PDEs in data generation. In addition to accurate reconstructions, the model is able to produce multiple plausible solutions for non-identifiable systems. Second, the thesis proposes a neural network architecture that mimics the structure of a PDE solver. The proposed architecture meets low data requirements of industrial settings and shows great potential for monitoring real-world chemical reactors. Third, the thesis presents novel algorithms for solving PDEs with neural networks by applying PDE information as a loss term during training. Unlike traditional methods, the developed algorithms incorporate available measurements and yield more accurate solutions with more stable solving procedures. Finally, the thesis introduces an algorithm based on pre-trained large language models to discover the analytical equations that govern the observed data. The results of the thesis demonstrate that neural networks can be effectively applied to physical system modeling with a handful of measurements through effective utilization of prior knowledge. In a broader context, this research opens new opportunities for the successful application of datadriven models in scientific and industrial contexts.