UC San Diego

This dissertation develops a data-driven framework for discovering steady-state structures in parameterized dynamical systems. Motivated by applications in which governing equations may be unavailable, incomplete, or costly to solve repeatedly, we formulate equilibrium discovery as the problem of learning parameter-dependent structures directly from observational data. In particular, the relation between parameters and steady-state solutions may be multivalued and may change across parameter regimes. We introduce a scalar target function Φ(U,Θ) defined on the joint solution–parameter space. For each fixed parameter value Θ, this target function is constructed as a Gaussian-mixture landscape whose peaks encode the existence, multiplicity, and locations of steady-state solutions. To approximate this landscape, we develop the Parameter–Solution Neural Network (PSNN), an architecture designed to capture the joint but decomposable dependence on solution and parameter variables. We establish a universal approximation theorem with explicit error bounds for the PSNN, including target functions with different regularity in the solution and parameter directions. Building on the learned landscape, we develop a computational framework for recovering steady-state structure from the PSNN-predicted landscape. The framework includes a cutoff-and clustering-based algorithm for locating equilibrium, auxiliary classification models for predicting equilibrium counts and stability, and adaptive refinement strategies for resolving closely spaced steady states. These components allow the method to recover solution multiplicity, locations, and stability information even in challenging regimes where equilibria are close to one another or only partially observed. Numerical experiments on the Gray–Scott model and two-gene feedback loop systems demonstrate that the proposed methods accurately recover multiple steady states, phase boundaries, bifurcation behavior, and stability structure. The results further show that the framework remains robust under incomplete observational data, supporting its use as a data-driven approach for equilibrium discovery in nonlinear parameterized dynamical systems.