Gaussian Processes simplify differential equations
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1.
California Institute of Technology
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2.
The Alan Turing Institute
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3.
Johns Hopkins University
Abstract
In this paper we use Gaussian processes (kernel methods) to learn mappings between trajectories of distinct differential equations. Our goal is to simplify both the representation and the solution of these equations. We begin by examining the Cole-Hopf transformation, a classical result that converts the nonlinear, viscous Burgers' equation into the linear heat equation. We demonstrate that this transformation can be effectively learned using Gaussian process regression, either from single or from multiple initial conditions of the Burgers equation. We then extend our methodology to discover mappings between initial conditions of a nonlinear partial differential equation (PDE) and a linear PDE, where the exact form of the linear PDE remains unknown and is inferred through Computational Graph Completion (CGC), a generalization of Gaussian Process Regression from approximating single input/output functions to approximating multiple input/output functions that interact within a computational graph. Further, we employ CGC to identify a local transformation from the nonlinear ordinary differential equation (ODE) of the Brusselator to its Poincaré normal form, capturing the dynamics around a Hopf bifurcation. Moreover, we interpret our learning procedure through Algorithmic Information Theory (AIT) and the Minimal Description Length (MDL) principle, framing these transformations as efficient, succinct encodings that compress nonlinear dynamics into simpler, linearized representations. This MDL perspective not only provides a theoretical justification for kernel-based regression methods but also illuminates the relationship between kernel learning and principles of model simplicity and data compression showing that learning in a reproducing kernel Hilbert space (RKHS) simultaneously minimizes a proxy for Kolmogorov complexity and maximizes algorithmic mutual information between the data and transformation. We conclude by addressing the broader question of whether systematic transformations between nonlinear and linear PDEs can generally exist, suggesting avenues for future research.
Copyright and License
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Acknowledgement
BH and YK gratefully acknowledge partial support from the Air Force Office of Scientific Research, United States (award number FA9550-21-1-0317) and the Department of Energy (award number SA22-0052-S001). HO gratefully acknowledges partial support from the Air Force Office of Scientific Research, United States under MURI award number FA9550-20-1-0358 (Machine Learning and Physics-Based Modeling and Simulation), from Beyond Limits (Computing Optimal Models), through the JPL Research and Technology Development program award (UQ-aware Machine Learning for Uncertainty Quantification) and by the Department of Energy under award number DE-SC0023163 (SEA-CROGS: Scalable, Efficient and Accelerated Causal Reasoning Operators, Graphs and Spikes for Earth and Embedded Systems). HO and BH acknowledge support from JPL/NASA under the AIST award “Kernel Flows: Emulating Complex Models for Massive Data Sets”. H.O. is grateful for the Department of Defense Vannevar Bush Faculty Fellowship.
Conflict of Interest
The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Boumediene Hamzi reports financial support was provided by Air Force Office of Scientific Research. Yannis Kevrekidis reports financial support was provided by Air Force Office of Scientific Research. Boumediene Hamzi reports financial support was provided by Department of Energy. Yannis Kevrekidis reports financial support was provided by Department of Energy. Houman Owhadi reports financial support was provided by Air Force Office of Scientific Research. Houman Owhadi reports financial support was provided by Beyond Limits. Houman Owhadi reports financial support was provided by Jet Propulsion Laboratory. Houman Owhadi reports financial support was provided by Department of Energy. Houman Owhadi reports financial support was provided by NASA. Boumediene Hamzi reports financial support was provided by NASA. Boumediene Hamzi is co-editor of Physica D. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Data Availability
The code for this paper can be found at www.github.com/jonghyeon1998/CGC.
Additional details
Funding
- United States Department of Energy
- SA22-0052-S001
- United States Air Force Office of Scientific Research
- FA9550-21-1-0317
- Jet Propulsion Laboratory
- AIST award “Kernel Flows: Emulating Complex Models for Massive Data Sets”
- United States Department of Defense
- SEA-CROGS: Scalable, Efficient and Accelerated Causal Reasoning Operators, Graphs and Spikes for Earth and Embedded Systems DE-SC0023163
- United States Air Force Office of Scientific Research
- MURI Machine Learning and Physics-Based Modeling and Simulation FA9550-20-1-0358
Caltech Custom Metadata
- Caltech groups
- Division of Engineering and Applied Science (EAS)
- Publication Status
- Published