[Paper] High-Dimensional Surrogate Modeling for Closed-Loop Learning of Neural-Network-Parameterized Model Predictive Control

Source: arXiv - 2512.11705v1

Overview

The paper tackles a practical bottleneck in data‑driven controller tuning: optimizing high‑dimensional controller parameters from closed‑loop experiments. By swapping the traditional Gaussian‑process (GP) surrogate used in Bayesian optimization (BO) for Bayesian neural network (BNN) surrogates, the authors demonstrate faster, more reliable learning even when the parameter space runs into the hundreds or thousands of dimensions—common in modern Model Predictive Control (MPC) setups.

Key Contributions

• Identifies the scalability limits of GP‑based BO for dense, high‑dimensional MPC parameterizations.
• Proposes Bayesian neural networks (both finite‑width and infinite‑width) as surrogate models for BO in control‑parameter learning.
• Empirical comparison on a cart‑pole benchmark showing BNN surrogates converge in fewer experiments and remain stable up to >1,000 parameters, where GPs break down.
• Provides practical guidelines for selecting surrogate models based on problem dimensionality and computational budget.

Methodology

1. Closed‑loop learning loop – The controller (an MPC with many tunable weights) runs on a simulated plant. After each experiment, the closed‑loop cost (e.g., tracking error + control effort) is measured.
2. Bayesian Optimization (BO) – BO builds a probabilistic surrogate of the cost‑to‑parameter mapping and uses an acquisition function (e.g., Expected Improvement) to propose the next parameter set.
3. Surrogate candidates
   • Gaussian Process (GP) with Matérn kernel – the classic BO surrogate.
   • Finite‑width Bayesian Neural Network (BNN) – a neural net with a modest number of hidden units, trained with variational inference to retain uncertainty estimates.
   • Infinite‑width Bayesian Neural Network (Neural‑Tangents GP) – leverages the Neural Tangent Kernel (NTK) to obtain a GP‑like model that scales better with dimensionality.
4. Benchmark task – A classic cart‑pole swing‑up problem where the MPC’s cost matrix and horizon weights are encoded in a vector of up to 1,200 dimensions.
5. Evaluation – Track the closed‑loop cost versus the number of BO iterations and compare convergence speed, final cost, and robustness across random seeds.

Results & Findings

Key takeaways

• BNN surrogates dramatically reduce the number of required experiments (≈ 30 % fewer than GP for 500‑dim problems).
• Infinite‑width BNNs maintain performance even when the parameter vector exceeds 1 k dimensions, a regime where GPs become ineffective.
• The surrogate’s ability to model complex, high‑dimensional landscapes translates directly into lower closed‑loop cost after fewer tuning cycles.

Practical Implications

• MPC developers can now automate tuning of large weight matrices (e.g., stage‑cost, terminal cost, soft constraints) without hand‑crafting low‑dimensional parameterizations.
• Reduced experimental budget – In industry, each closed‑loop trial may involve costly hardware or long simulation runs; BNN‑based BO can cut that budget by a third to a half.
• Scalable to modern embedded AI‑control stacks where controllers are parameterized by deep neural nets or large linear‑quadratic regulators.
• Toolchain integration – The approach fits naturally into existing Python‑based BO libraries (e.g., BoTorch, GPyTorch) by swapping the surrogate class, making adoption low‑friction.
• Safety‑critical domains (automotive, aerospace) benefit from the probabilistic nature of BNNs, which still provide uncertainty estimates for safe exploration.

Limitations & Future Work

• Computational overhead – Training BNNs (especially finite‑width) adds per‑iteration cost; while still cheaper than many experiments, real‑time tuning on ultra‑resource‑constrained hardware may need further optimization.
• Benchmark scope – The study focuses on a single cart‑pole task; broader validation on higher‑order plants (e.g., multi‑link manipulators, power‑grid frequency control) is needed.
• Acquisition function tuning – The paper uses standard Expected Improvement; exploring more robust acquisition strategies for very high dimensions could yield additional gains.
• Robustness to noise – Real‑world measurements often contain sensor noise and disturbances; extending the analysis to noisy cost signals is a natural next step.

Bottom line: By replacing Gaussian‑process surrogates with Bayesian neural networks, the authors open the door to sample‑efficient, high‑dimensional controller learning, a capability that aligns well with the growing complexity of modern MPC and AI‑augmented control systems.

Authors

• Sebastian Hirt
• Valentinus Suwanto
• Hendrik Alsmeier
• Maik Pfefferkorn
• Rolf Findeisen

Paper Information

• arXiv ID: 2512.11705v1
• Categories: cs.LG, eess.SY
• Published: December 12, 2025
• PDF: Download PDF