A Schrödinger Eigenfunction Method for Long-Horizon Stochastic Optimal Control

2026-03-24

AI Summary

This work addresses the challenges of high-dimensional stochastic optimal control over long planning horizons, where computational complexity typically grows linearly with time and performance degrades. Focusing on linearly solvable control problems with gradient-drift structure, the authors transform the Hamilton–Jacobi–Bellman equation into a linear partial differential equation governed by an operator ℒ, and establish the unitary equivalence between ℒ and a Schrödinger operator. Leveraging this insight, they derive—for the first time—an analytical solution for symmetric linear quadratic regulators (LQR) with arbitrary terminal costs. By integrating spectral methods, analytical solutions from quantum harmonic oscillators, and neural networks, they design a novel loss function to efficiently learn the eigenfunctions of ℒ. The resulting approach achieves an order-of-magnitude improvement in control accuracy over existing methods on multiple long-horizon benchmark tasks, while reducing both memory and computational complexity from 𝒪(Td) to 𝒪(d).

Technology Category

• Search and Optimization: Learning to Search
• Intelligent Robots: Learning & Optimization for ROB
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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 Schrödinger operator $\mathcal{S} = -Δ+ \mathcal{V}$ 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 $\mathcal{O}(d)$.

Problem

• stochastic optimal control
• long-horizon planning
• high-dimensional control
• Hamilton-Jacobi-Bellman equation
• eigenfunction approximation

Innovation

• Schrödinger eigenfunction
• stochastic optimal control
• Hamilton-Jacobi-Bellman equation
• neural eigenfunction learning
• long-horizon control