UC Santa Barbara

Machine learning (ML) techniques have emerged as powerful tools for solving differential equations, particularly in the context of partial differential equations (PDEs), enabling accelerated forward simulations and parameter discovery from limited data.However, challenges persist in maintaining numerical accuracy, especially in scenarios requiring real-time inference and inverse problem-solving. This dissertation investigates innovative hybrid strategies that blend classical finite discretization methods with modern ML techniques to enhance accuracy while maintaining computational efficiency. Our research focuses on addressing key challenges at the intersection of scientific computing, machine learning, and applied mathematics, including performance, accuracy, and data efficiency.