Physics-Informed Kernel Embeddings: Integrating Prior System Knowledge with Data-Driven Control
Physics-informed kernel embeddings incorporate prior system knowledge, improving sample efficiency and prediction accuracy in control tasks.
Key Findings
Methodology
This paper introduces a physics-informed kernel embedding framework based on RKHS, where prior system dynamics are embedded as a bias term in a regularized least squares problem. The method leverages kernel functions k and l over state and control spaces, with a bias function f0 representing approximate dynamics. The resulting optimization admits a closed-form solution, enabling efficient state prediction and control policy synthesis. The approach combines data-driven observations with physics priors, leading to improved sample efficiency and generalization, especially in systems with limited data or model inaccuracies.
Key Results
- In spring-mass-damper systems, the physics-informed kernel embedding reduces prediction errors by over 90% compared to purely data-driven models with only 10-50 samples, maintaining high accuracy even in limited data regimes.
- For the F-16 aircraft model, with 500 samples, the method achieves about 70% lower prediction error than baseline models, accurately capturing nonlinear dynamics and control responses.
- In target tracking with nonholonomic vehicles, the physics-informed control policy outperforms pure data-driven strategies, achieving closer trajectory tracking and 20-fold faster computation, demonstrating robustness and efficiency.
Significance
This work addresses the critical challenge of data scarcity in control systems by integrating physics knowledge into kernel methods. It enhances the reliability and generalization of data-driven models, reducing the need for large datasets and improving robustness against model misspecification. The approach bridges the gap between classical physics-based control and modern machine learning, opening pathways for deploying autonomous systems in real-world, data-limited environments with higher safety and efficiency.
Technical Contribution
The core innovation is embedding prior dynamics as a bias term within the kernel embedding framework, leading to a closed-form solution for the learned distribution. Theoretically, the paper proves the uniqueness and stability of the solution, and practically, it enables fast linear programming-based control policy computation. This hybrid approach combines the strengths of physics-based modeling and nonparametric learning, offering a new tool for scalable, robust control design.
Novelty
This is the first work to directly incorporate imperfect prior system knowledge as a bias in kernel distribution embeddings, rather than modifying kernels or relying solely on neural networks. The method provides a principled, closed-form solution that enhances data efficiency and generalization, distinguishing it from existing kernel or physics-informed neural network approaches that often require extensive training or lack theoretical guarantees.
Limitations
- The effectiveness depends on the quality of the prior knowledge; large deviations in the approximate dynamics can degrade performance.
- Kernel computations become computationally intensive in high-dimensional spaces, limiting scalability.
- The method assumes static system dynamics; adapting to time-varying or highly nonlinear systems remains a challenge, requiring further development.
Future Work
Future research will focus on adaptive bias functions that can learn and update prior knowledge online, extending the framework to continuous-time and partially observed systems. Integrating deep kernel techniques and reinforcement learning strategies could further improve performance in complex, high-dimensional environments, making the approach more versatile for real-time control applications.
AI Executive Summary
In autonomous control systems, the challenge of limited data and imperfect system models hampers reliable prediction and control. Traditional data-driven methods like kernel embeddings excel at capturing complex distributions but often require large datasets and neglect valuable prior knowledge about system physics. This paper introduces a novel physics-informed kernel embedding framework that embeds approximate system dynamics directly into the kernel learning process as a bias term. By doing so, it effectively combines observational data with prior system knowledge, significantly boosting sample efficiency and prediction accuracy.
The core technical innovation lies in formulating a regularized least squares problem with a bias term, which admits a closed-form solution. This enables efficient computation of the expected system evolution and control policies through linear programming. The approach was validated on multiple challenging tasks, including a spring-mass-damper system, an F-16 fighter aircraft model, and a target tracking problem with nonholonomic dynamics. In all cases, the physics-informed method outperformed purely data-driven models, especially with limited samples, reducing prediction errors by over 90% in some scenarios.
These results demonstrate that incorporating prior physics knowledge into kernel methods can dramatically improve the robustness and efficiency of data-driven control. The method’s theoretical guarantees and computational simplicity make it a promising tool for real-world applications where data is scarce or expensive to obtain. Looking ahead, the authors plan to extend the framework to adaptive, online learning settings and higher-dimensional systems, potentially revolutionizing how autonomous systems leverage physics priors for safer, more reliable operation.
Deep Dive
Abstract
Data-driven control algorithms use observations of system dynamics to construct an implicit model for the purpose of control. However, in practice, data-driven techniques often require excessive sample sizes, which may be infeasible in real-world scenarios where only limited observations of the system are available. Furthermore, purely data-driven methods often neglect useful a priori knowledge, such as approximate models of the system dynamics. We present a method to incorporate such prior knowledge into data-driven control algorithms using kernel embeddings, a nonparametric machine learning technique based in the theory of reproducing kernel Hilbert spaces. Our proposed approach incorporates prior knowledge of the system dynamics as a bias term in the kernel learning problem. We formulate the biased learning problem as a least-squares problem with a regularization term that is informed by the dynamics, that has an efficiently computable, closed-form solution. Through numerical experiments, we empirically demonstrate the improved sample efficiency and out-of-sample generalization of our approach over a purely data-driven baseline. We demonstrate an application of our method to control through a target tracking problem with nonholonomic dynamics, and on spring-mass-damper and F-16 aircraft state prediction tasks.