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A Schrödinger Eigenfunction Method for Long-Horizon Stochastic Optimal Control

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
Stochastic Optimal ControlSchrödinger operatoreigenfunction learninglong-horizon control
Abstract:

High-dimensional stochastic optimal control (SOC) becomes harder with longer planning horizons: existing methods scale linearly in the horizon TT, 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 L\mathcal{L}. We prove that, under the gradient drift assumption, L\mathcal{L} is unitarily equivalent to a Schrödinger operator S=−Δ+V\mathcal{S} = -\Delta + \mathcal{V} with purely discrete spectrum, allowing the long-horizon control to be efficiently described via the eigensystem of L\mathcal{L}. This connection provides two key results: first, for a symmetric linear-quadratic regulator (LQR), S\mathcal{S} matches the Hamiltonian of a quantum harmonic oscillator, whose closed-form eigensystem yields an analytic solution to the symmetric LQR with arbitrary terminal cost. Second, in a more general setting, we learn the eigensystem of L\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 O(Td)\mathcal{O}(Td) to O(d)\mathcal{O}(d).

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
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Overview

Overall Novelty Assessment

The paper introduces a spectral framework for long-horizon stochastic optimal control by establishing unitary equivalence between the HJB operator and a Schrödinger operator with discrete spectrum. It occupies the sole position in the 'Schrödinger Operator Eigenfunction Methods' leaf, which itself is the only leaf under 'Spectral and Eigenfunction-Based Methods'. This isolation in the taxonomy suggests the approach represents a relatively unexplored direction within the broader field of gradient-drift control, where most work concentrates on gradient-based optimization, dual control, or reinforcement learning branches.

The taxonomy reveals five neighboring branches addressing long-horizon control through different lenses. Theoretical Foundations establish optimality conditions for irregular drift, while Gradient-Based Optimization develops iterative PDE-constrained schemes like adaptive gradient descent. Dual Control balances parameter estimation with control objectives, and Drift Rate Control handles switching dynamics. The spectral approach diverges fundamentally from these directions by exploiting operator-theoretic structure rather than iterative updates or online adaptation, positioning it as a complementary analytical tool rather than a competing algorithmic framework.

Among twenty-two candidates examined via semantic search, none clearly refute any of the three core contributions. The Schrödinger operator framework examined ten candidates with zero refutations, the closed-form symmetric LQR solution examined ten with zero refutations, and the relative eigenfunction loss examined two with zero refutations. This absence of overlapping prior work within the limited search scope suggests the spectral reduction and eigenfunction learning approach may represent a genuinely distinct methodological contribution, though the modest candidate pool leaves open the possibility of relevant work outside the top-K semantic matches.

Based on the limited literature search covering twenty-two semantically related papers, the work appears to introduce novel operator-theoretic machinery not directly anticipated by existing gradient-based, dual control, or RL methods in the taxonomy. The analysis does not claim exhaustive coverage of all stochastic control literature, and the sparse population of the spectral methods branch may reflect either genuine novelty or incomplete taxonomy construction rather than definitive field-wide uniqueness.

Taxonomy

Core-task Taxonomy Papers
12
3
Claimed Contributions
22
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: long-horizon stochastic optimal control with gradient drift. The field addresses how to steer dynamical systems subject to random perturbations and drift forces over extended time horizons, balancing theoretical rigor with computational tractability. At the highest level, the taxonomy reveals six main branches. Theoretical Foundations and Maximum Principles establish rigorous conditions under which optimal controls exist, even when drift coefficients are irregular or merely measurable, as exemplified by Maximum Principle Measurable Drifts[2]. Spectral and Eigenfunction-Based Methods exploit operator-theoretic tools—particularly Schrödinger operator eigenfunctions—to characterize long-time behavior and optimal policies. Gradient-Based Optimization for Stochastic Control develops algorithmic schemes that adapt gradient descent to parabolic PDEs and stochastic settings, including works like Adaptive SGD Parabolic[5] and Adaptive Gradient Parabolic[7]. Dual Control and Parameter Estimation tackle scenarios where system parameters themselves drift or must be learned online, as in Parameter Drift Control[1]. Drift Rate Control and Impulse Control focus on problems with switching dynamics or abrupt interventions, such as Switched Drift Fault[4]. Finally, Reinforcement Learning and Model-Based Approaches bridge classical control with data-driven policy search, illustrated by World Models Generalization[3] and Policy Gradient Multiplicative Noise[10]. A particularly active contrast emerges between spectral methods, which offer elegant asymptotic characterizations but can be computationally demanding, and gradient-based or RL techniques that prioritize scalability and online adaptation. Within the spectral branch, Schrodinger Eigenfunction Control[0] leverages eigenfunction expansions to derive control laws for long-horizon problems with gradient drift, positioning itself close to the operator-theoretic tradition yet distinct from purely numerical gradient schemes like Adaptive SGD Parabolic[5]. While gradient-based methods emphasize iterative updates and convergence rates in high dimensions, Schrodinger Eigenfunction Control[0] exploits the structure of the underlying PDE to obtain more explicit representations of optimal feedback. This trade-off between analytical insight and computational flexibility remains a central open question, with some works exploring hybrid strategies that combine spectral decompositions with adaptive optimization.

Claimed Contributions

Schrödinger operator framework for gradient-drift stochastic optimal control

The authors establish that for stochastic optimal control problems where the uncontrolled drift is the gradient of a potential, the Hamilton-Jacobi-Bellman equation reduces to a linear PDE governed by an operator unitarily equivalent to a Schrödinger operator with discrete spectrum. This connection enables efficient long-horizon control via the operator's eigensystem.

10 retrieved papers
Closed-form solution for symmetric LQR with arbitrary terminal cost

For the special case of symmetric linear-quadratic regulator problems, the authors derive an analytic solution that removes the classical requirement of quadratic terminal cost by exploiting the equivalence between the associated Schrödinger operator and the quantum harmonic oscillator Hamiltonian.

10 retrieved papers
Relative eigenfunction loss for control-oriented learning

The authors introduce a relative eigenfunction loss that eliminates implicit spatial reweighting present in existing PINN and variational losses. This new loss remains sensitive in high-value-function regions, which are critical for control synthesis, addressing a key limitation of prior eigenfunction learning methods.

2 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Within the taxonomy built over the current TopK core-task papers, the original paper is assigned to a leaf with no direct siblings and no cousin branches under the same grandparent topic. In this retrieved landscape, it appears structurally isolated, which is one partial signal of novelty, but still constrained by search coverage and taxonomy granularity.

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Schrödinger operator framework for gradient-drift stochastic optimal control

The authors establish that for stochastic optimal control problems where the uncontrolled drift is the gradient of a potential, the Hamilton-Jacobi-Bellman equation reduces to a linear PDE governed by an operator unitarily equivalent to a Schrödinger operator with discrete spectrum. This connection enables efficient long-horizon control via the operator's eigensystem.

Contribution

Closed-form solution for symmetric LQR with arbitrary terminal cost

For the special case of symmetric linear-quadratic regulator problems, the authors derive an analytic solution that removes the classical requirement of quadratic terminal cost by exploiting the equivalence between the associated Schrödinger operator and the quantum harmonic oscillator Hamiltonian.

Contribution

Relative eigenfunction loss for control-oriented learning

The authors introduce a relative eigenfunction loss that eliminates implicit spatial reweighting present in existing PINN and variational losses. This new loss remains sensitive in high-value-function regions, which are critical for control synthesis, addressing a key limitation of prior eigenfunction learning methods.

Table of Contents

A Schrödinger Eigenfunction Method for Long-Horizon Stochastic Optimal Control | Novelty Validation