Learning To Solve Differential Equation Constrained Optimization Problems

TL;DR

This work addresses the computational bottlenecks of solving DE-constrained optimization problems by introducing DE-OP, a dual-network framework that learns both decision policies and system dynamics in a differentiable, end-to-end manner. A proxy optimizer $\mathcal{F}_{\omega}$ estimates near-optimal controls, while a neural-DE model $\mathcal{N}_{\theta}$ predicts state trajectories, with training guided by a primal-dual augmented-Lagrangian loss to enforce both static and dynamic constraints. Across dynamic portfolio optimization and stability-constrained AC-OPF, DE-OP achieves high-fidelity dynamic compliance and markedly improved decision quality—up to 25x better than static proxies—and operates in near real time. The results demonstrate the practical potential of integrating learned dynamics into optimization workflows for complex, real-time decision-making in energy, finance, and beyond.

Abstract

Differential equations (DE) constrained optimization plays a critical role in numerous scientific and engineering fields, including energy systems, aerospace engineering, ecology, and finance, where optimal configurations or control strategies must be determined for systems governed by ordinary or stochastic differential equations. Despite its significance, the computational challenges associated with these problems have limited their practical use. To address these limitations, this paper introduces a learning-based approach to DE-constrained optimization that combines techniques from proxy optimization and neural differential equations. The proposed approach uses a dual-network architecture, with one approximating the control strategies, focusing on steady-state constraints, and another solving the associated DEs. This combination enables the approximation of optimal strategies while accounting for dynamic constraints in near real-time. Experiments across problems in energy optimization and finance modeling show that this method provides full compliance with dynamic constraints and it produces results up to 25 times more precise than other methods which do not explicitly model the system's dynamic equations.

Figures (13)

• Figure 1: Decision variables ${\textcolor{blue}{u}}$ represent generators outputs, which are influenced by state variables ${\textcolor{brown}{y}}$ describing rotor angles and speed.
• Figure 2: DE-OP uses a dual network architecture consisting of a proxy optimization model $\mathcal{F}_\omega$ to estimate the decision variables $\hat{\bm u}$ and a neural-DE model $\mathcal{N}_\theta$ to estimate the state-variables $\hat{\bm y}(t)$, with the objective function $\mathcal{J}(\mathcal{F}_\omega(\bm \zeta), \mathcal{N}_\theta \left(\mathcal{F}_\omega(\bm \zeta), t \right); \bm{\zeta} )$ capturing the overall loss.
• Figure 3: Average Opt. gap with $n=50$ asset prices.
• Figure 4: WSCC $9$-bus - Percentage of unstable dynamics observed during training. Average of 40 experiments.
• Figure 5: IEEE $57$-bus - Percentage of unstable dynamics observed during training. Average of 40 experiments.