UC Merced

Time-evolving systems arise throughout science and engineering whenever a governing equation is used to describe how a state evolves in time. A central goal in these settings is to accurately simulate, forecast, or infer this evolution, but practical progress is often limited by a combination of computational and data constraints. Common obstacles include expensive multiscale discretizations, temporal nonlocality induced by memory terms, and limited observations available for calibration. These difficulties are especially acute in quantum and electron dynamics, where relevant state spaces grow exponentially and near-term quantum hardware yields imperfect, noisy measurements. To this end, this dissertation develops equation-aware, data-driven methods that bring expensive, often intractable, computations within practical reach while preserving accuracy on targeted objectives. Specifically, the contributions are threefold.(1) For integro-differential equations (IDEs) motivated by the Kadanoff–Baym equations, explicit history integrals force numerical solvers to scale quadratically in the number of time steps. Using an analytic result from many-body perturbation theory that implies a deterministic map from the instantaneous Green’s function to the history integral, we replace the history integral with a time-local update learned by a neural network and obtain linear-in-time cost at target accuracy.(2) For spatiotemporal systems with only short paired coarse–fine observation windows, we propose a coarse-to-fine super-resolution and forecasting framework that couples a low-cost coarse integrator with a learned super-resolution operator to emulate fine-grid evolution without fine-grid time stepping. We evaluate the approach on canonical partial differential equations (2D heat, wave, and incompressible Navier–Stokes) and a reduced Vlasov–Poisson system for electron motion in a plasma, quantify accuracy versus compute/storage, and provide a novel error analysis for long-horizon error accumulation.(3) For ground-state energy estimation of molecular Hamiltonians from noisy time-evolution signals acquired on near-term quantum hardware, we introduce a noise-robust post-processing algorithm that achieves chemical accuracy with reduced quantum overhead and reduced sensitivity to hyperparameters.Taken together, (1)–(3) show that targeting computational bottlenecks in systems with modular data-driven components, guided by the equations of motion and the measurement process, can substantially improve efficiency and robustness, whilst also maintaining accuracy on the targeted simulation and inference objectives.