A Schr\"odinger Eigenfunction Method for Long-Horizon Stochastic Optimal Control

TL;DR

This paper introduces a Schr"odinger eigenfunction approach to solve long-horizon stochastic optimal control problems efficiently, leveraging spectral theory and neural networks for improved accuracy and scalability.

Contribution

It establishes a novel connection between SOC and Schr"odinger operators, providing analytic solutions for LQR and neural network-based eigensystem learning for general cases.

Findings

Achieved an order-of-magnitude improvement in control accuracy.

Reduced memory and runtime complexity from O(Td) to O(d).

Provided analytic solutions for symmetric LQR with arbitrary terminal costs.

Abstract

High-dimensional stochastic optimal control (SOC) becomes harder with longer planning horizons: existing methods scale linearly in the horizon $T$, with performance often deteriorating exponentially. We overcome these limitations for a subclass of linearly-solvable SOC problems-those whose uncontrolled drift is the gradient of a potential. In this setting, the Hamilton-Jacobi-Bellman equation reduces to a linear PDE governed by an operator $\mathcal{L}$. We prove that, under the gradient drift assumption, $\mathcal{L}$ is unitarily equivalent to a Schr\"odinger operator $\mathcal{S} = -\Delta + \mathcal{V}$ with purely discrete spectrum, allowing the long-horizon control to be efficiently described via the eigensystem of $\mathcal{L}$. This connection provides two key results: first, for a symmetric linear-quadratic regulator (LQR), $\mathcal{S}$ matches the Hamiltonian of a quantum harmonic oscillator, whose closed-form eigensystem yields an analytic solution to the symmetric LQR with \emph{arbitrary} terminal cost. Second, in a more general setting, we learn the eigensystem of $\mathcal{L}$ using neural networks. We identify implicit reweighting issues with existing eigenfunction learning losses that degrade performance in control tasks, and propose a novel loss function to mitigate this. We evaluate our method on several long-horizon benchmarks, achieving an order-of-magnitude improvement in control accuracy compared to state-of-the-art methods, while reducing memory usage and runtime complexity from $\mathcal{O}(Td)$ to $\mathcal{O}(d)$.