Learning dominant physical processes with data-driven balance models (2001.10019v2)

Abstract: Throughout the history of science, physics-based modeling has relied on judiciously approximating observed dynamics as a balance between a few dominant processes. However, this traditional approach is mathematically cumbersome and only applies in asymptotic regimes where there is a strict separation of scales in the physics. Here, we automate and generalize this approach to non-asymptotic regimes by introducing the idea of an equation space, in which different local balances appear as distinct subspace clusters. Unsupervised learning can then automatically identify regions where groups of terms may be neglected. We show that our data-driven balance models successfully delineate dominant balance physics in a much richer class of systems. In particular, this approach uncovers key mechanistic models in turbulence, combustion, nonlinear optics, geophysical fluids, and neuroscience.

• The paper introduces a data-driven balance modeling method using unsupervised learning to identify dominant physical processes in complex systems.
• Mapping equation terms into an 'equation space', unsupervised clustering identifies dominant balance relationships as geometric clusters within the space.
• Applied to systems like fluid flows and optics, the method successfully identifies physically interpretable regions consistent with classical analysis and scaling laws.

Overview of Data-Driven Balance Models for Identifying Dominant Physical Processes

This paper presents a novel method for identifying dominant physical processes across a range of complex systems via a data-driven approach known as balance modeling. The authors propose unsupervised learning algorithms that automatically discern key mechanistic models by segregating different local balance regimes into distinguishable clusters within an equation space. This approach differs from traditional physics-based modeling, which often relies on asymptotic approximations that necessitate a strict separation of scales. Instead, this method generalizes such approximations and extends them beyond asymptotic regimes, offering broader applicability.

Key Contributions

The primary contribution of this research lies in introducing an automated method capable of delineating dominant physics in a wide array of physical systems. The foundation of this approach rests on:

• Equation Space Framework: The authors define an 'equation space' where each term in a governing equation is a coordinate. This space allows for a geometric interpretation of dominant balance physics, characterized by regions where certain terms can be neglected. This approach reveals local balance relations as clusters within this multidimensional space.
• Unsupervised Learning for Clustering: The utilization of Gaussian mixture models (GMM) to cluster data in equation space, subsequently using sparse principal component analysis (SPCA) to extract significant directions of variance characterized by subsets of active terms.
• Physical Applications: The methodology is applied to various complex systems including fluid flows (turbulent boundary layers, vortex shedding), optical pulse propagation, neuroscience (Hodgkin-Huxley neuron models), and rotating detonation engines. Each case paper demonstrates the method's effectiveness in identifying balance models consistent with classical analysis and physical intuition.

Results and Implications

Quantitatively, the method successfully segmented complex fields into physically interpretable regions consistent with known theoretical models. For example, in the case of turbulent boundary layers, the method identified well-established layers such as the viscous sublayer and inertial sublayer, correlating with classical scaling laws and the "law of the wall". Similarly, in optical fiber simulations, regions of soliton propagation were associated with lower-order dispersion terms balanced by nonlinear effects. These results affirm the versatility and correctness of the approach across non-trivial physical systems.

The implications of this method are two-fold. Practically, it eliminates the need for manual parameter tuning or expert intervention in identifying dominant balance relations, allowing the method to be applied broadly and efficiently to novel systems or those traditionally resistant to asymptotic analysis. Theoretically, it provides a systematic framework to embolden traditional approaches like scaling analyses while offering new insights into systems with multiscale dynamics.

Future Directions

Future developments could explore the integration of real-time data streams to dynamically adjust the model in changing environments, thus contributing to predictive modeling and control. Further expansion into domains such as biological systems, meteorology, or materials science could open new research avenues, as these often lack the deterministic equations characteristic of physics-based systems.

In summary, this paper delivers substantial advancements in identifying dominant physical processes using unsupervised data-driven methods, offering promising applications across diverse scientific and engineering domains.