This dissertation develops a data-driven framework for equilibrium discovery in parameterized dynamical systems. The central idea is to represent the solution set through a parameter-solution neural network (PSNN), which learns a scalar landscape Φ(u,θ) on the product space of state variables and parameters. Peaks of this learned landscape encode steady states, allowing the method to recover varying numbers of equilibria across parameter regimes. The dissertation further develops adaptive refinement procedures and classifier-assisted algorithms to improve the localization of closely spaced solutions, infer stability, and remain robust under incomplete observations. Finally, it establishes approximation-theoretic guarantees for the PSNN architecture and demonstrates the framework on benchmark dynamical and biochemical reaction systems.