Certifiable Neural Operators for Simultaneously Solving Infinitely Many Convex Programs: Logarithmic Rank and Linear Complexity

This arXiv paper proposes proximal equilibrium operators (PEOs), constructed via finite-dimensional deep equilibrium layers, that can solve a family of convex optimization problems on separable Hilbert spaces. The paper proves that when input loss functions lie in a suitable infinite-dimensional set, PEOs can uniformly approximate corresponding solutions to arbitrary precision, with rank and width growing only logarithmically with the inverse of approximation error, and depth growing linearly. The related theory and trainability have been validated across three application domains: nonlinear PDEs, stochastic optimal control, and financial portfolio optimization and hedging.