Computational modeling of physical systems has become indispensable to modern engineering practice, enabling iterative virtual design before physical prototyping. This proposal identifies and addresses two fundamental bottlenecks in the computer-aided engineering workflow: the simulation bottleneck, arising from the poor computational scaling of classical methods, and the optimization bottleneck, stemming from the absence of gradient feedback through the simulation loop that forces engineers to rely on expensive zero-order design exploration.

To address these bottlenecks, this work develops simulation techniques that are simultaneously more computationally efficient and differentiable, enabling gradient-based design optimization. Contributions span four open problem categories: simulation, data representation, data generation, and performance evaluation. On the simulation front, we accelerate the shooting and bouncing ray
algorithm to achieve constant memory usage with improved runtime, then extend it to near-field optics for end-to-end diffractive optical element design. Bridging simulation and representation, we investigate neural networks as basis elements within finite element solvers. In later work, we provide a theoretical analysis of multigrid parametric encodings common in implicit neural representations via neural tangent kernel theory.

Looking forward, this proposal proposes two directions targeting the data generation and evaluation gaps that currently limit neural operator adoption in engineering. The first is a large-scale benchmarking suite spanning complex, irregular geometries in electromagnetics, fluid dynamics, and acoustics, with evaluation protocols that characterize how accuracy scales with model parameters and dataset size. The second is a synthetic data generation framework based on adversarial training and learned coordinate warping that reduces dependence on expensive classical solvers during neural operator training. Together, these contributions advance both the theoretical foundations and practical tooling needed to make differentiable, neural operator-based simulation a viable component of real-world engineering workflows.