Seminars

Traditional numerical methods for solving partial differential equations (PDEs) governing physical phenomena require significant computational resources. For inverse problems, the cost is further amplified, as traditional approaches, such as the Tikhonov regularization framework, require hundreds or thousands of forward solves during the optimization process. It is not mentioned that tuning the regularization strength is often an ad hoc and problem-dependent process, requiring extra effort to identify the optimal values. Therefore, conventional numerical methods often fail to meet the real-time solution requirements of various modern engineering applications. Recent advancements in machine learning surrogate models have shown promise in addressing these computational challenges. However, training surrogate models requires an extensive amount of data and often lacks physical interpretability and generalizability, especially in scenarios with limited data or when extrapolating beyond the training distribution. On the other hand, this is precisely the issue in many engineering applications, where data is often limited and sometimes costly to acquire. The dissertation addresses these challenges by developing novel model-constrained machine learning approaches that pave the way for real-time forward and inverse surrogate solvers while maintaining physical fidelity and accuracy.