Learning dominant physical processes with data-driven balance models

The study investigates how to automatically identify dominant physical processes that govern local behavior in complex systems without relying on asymptotic scale separation. Dominant balance heuristics have historically yielded reduced-order mechanistic models in fields such as turbulence, geophysical fluid dynamics, and fiber optics by identifying balances among a subset of terms in governing equations. Classical practice uses nondimensional parameters (e.g., Reynolds, Rossby, Froude, Rayleigh numbers) to determine important mechanisms and neglect others, but is limited to specific asymptotic regimes and lacks pointwise, local delineation in complex geometries. The research question is whether data-driven methods, linked directly to governing equations, can objectively and locally identify regimes where subsets of terms dominate, thus generalizing dominant balance analysis beyond strict asymptotics. The authors introduce an equation space representation in which each term in the governing equation defines a coordinate, so local balances manifest as clusters with low variance along negligible-term directions. This geometric perspective enables unsupervised learning to detect dominant processes and segment spatiotemporal domains accordingly.

Dominant balance has been central across physics and engineering, notably in boundary-layer theory and large-scale geophysical flows, where scaling analyses motivate simplified models (e.g., geostrophy, Ekman layers, thermal wind). Prior data-driven efforts have applied clustering with expert-crafted features or interpreted clusters post hoc in terms of balances, but a general, direct identification from data remained open. Concurrently, model discovery frameworks (e.g., sparse identification of nonlinear dynamics and PDE discovery) provide governing equations from data but do not explicitly partition local balance regimes. This work connects modern unsupervised learning with classical asymptotic insights by operating in equation space to discover local balances objectively.

Core idea: Represent the governing evolution equation in an implicit form N(u)=Σ_{i=1}^K f_i(u, derivatives,...)=0 and define an equation space whose coordinates are the individual terms f_i. For each spatiotemporal sample (x,t), evaluate all terms to obtain a K-dimensional vector f(x,t). In a dominant balance regime, only p<K terms are active and the remaining terms are near zero, so data concentrate near a p-dimensional subspace aligned with active-term axes.

Algorithmic pipeline:

• Equation space embedding: From measured or simulated fields u(x,t), compute all required spatial/temporal derivatives and evaluate each term f_i to form f(x,t) for all samples.
• Unsupervised clustering with Gaussian mixture models (GMMs): Fit a probabilistic mixture model to the set of equation-space vectors. Clusters corresponding to dominant balances exhibit small variance along negligible-term directions and larger variance along active-term directions. The GMM provides soft assignments and uncertainty estimates compatible with the relative nature of balance analysis.
• Sparse principal component analysis (SPCA): Because real data need not be Gaussian and clusters may be redundant or ambiguous, apply SPCA within each GMM cluster to obtain a sparse approximation to the leading principal component (via ℓ1 regularization). Nonzero entries of this sparse vector indicate active terms; near-zero entries indicate negligible terms.
• Regime aggregation: Group clusters sharing the same SPCA sparsity pattern into a single dominant balance model. Map these models back to the original domain to segment space-time into locally active physics.

Properties and advantages:

• No need to assume asymptotic scaling; yields pointwise, local estimates in complex geometries.
• Direct interpretability: equation-space axes map to physical processes.
• Probabilistic framework enables uncertainty quantification. The approach complements, not replaces, classical analysis.

Implementation details: Methods formalized as Eq. (7)-(8) for equation-space construction, followed by standard GMM training and SPCA within clusters to identify active terms. Data and code resources are provided for reproducibility.

Across five disparate systems, the method identifies physically interpretable dominant balance regimes consistent with theory and known behavior:

• Transitional turbulent boundary layer: From Reynolds-averaged Navier–Stokes terms, clusters reveal laminar inflow, viscous sublayer, inertial sublayer, slightly perturbed free stream, and a transitional region. The wall-normal extent of the inertial sublayer grows approximately as x^0.81, close to the theoretical x^(4/5) scaling from boundary-layer theory.
• Nonlinear optical pulse propagation (GNLSE, supercontinuum generation): Most of the field is well-described by linear dispersion (various orders). The strongest soliton region is identified with a balance between the cubic Kerr nonlinearity (instantaneous Raman delta component) and dispersive terms up to fourth order; the standard NLS (cubic + second-order dispersion only) is not selected. The full Raman time-delayed response is not identified as dominant for the soliton, indicating limited GMM sensitivity to that effect in this dataset and suggesting fourth-order dispersion significantly impacts emergent solitons (consistent with pure-quartic soliton observations).
• Geostrophic balance in the Gulf of Mexico (HYCOM surface currents): Three regimes are identified: (1) geostrophic balance (Coriolis balancing pressure gradient) in slowly varying, large-scale currents (e.g., southern Gulf Stream, Cuba–Yucatan channel), (2) acceleration–pressure-gradient balance (time-varying regions), and (3) linearized rotating Navier–Stokes. Nonlinear advection is not included in any selected model, consistent with linear wave dynamics in these regions.
• Generalized Hodgkin–Huxley neuron (Aplysia R15 bursting): During quiescent/slow oscillations, balances are dominated by calcium-dependent currents (ICaP, ISI, INaCa). During spikes, voltage-gated currents dominate: rising phase via inward sodium (INa), then peak and repolarization via delayed rectifier potassium (IK), matching known sodium–potassium spiking biophysics.
• Rotating detonation engine analog (Burgers–Majda surrogate): Four regions are detected in a wave-attached frame with two traveling waves: (i) shock front governed by Burgers-type balance (nonlinear advection–dissipation; kinetics negligible), (ii) reaction onset where energy input and nonlinear dissipation balance as kinetics activate, (iii) refractory region where energy input is negligible (λ≈1) but dissipation remains significant, and (iv) background region with Burgers dynamics balanced with autocatalytic energy input. These regimes align with known qualitative RDE dynamics (wave nucleation, modulation, mode-locking).

The equation-space geometric perspective ties unsupervised learning directly to governing physics: dominant balances appear as clusters with sparse covariance, enabling automated, interpretable identification of active processes. The GMM-SPCA framework recovers classical scaling results (boundary layer, geostrophy) and quantitatively confirms heuristic physical interpretations (nonlinear optics soliton structure, Hodgkin–Huxley bursting balances, RDE regimes). Compared with traditional asymptotic scaling, the approach provides pointwise, local regime identification without requiring strict scale separation and accommodates arbitrarily complex geometries while offering uncertainty estimates. The method is designed to complement, not replace, physical expertise and classical analysis, providing a principled and reproducible tool for testing hypothesized balances and guiding reduced-order mechanistic modeling.

This work introduces data-driven balance models based on an equation-space embedding combined with GMM clustering and SPCA to learn local dominant physical processes from data. The approach generalizes classical dominant balance analysis beyond asymptotic regimes, produces interpretable local models, and successfully delineates regimes across turbulence, nonlinear optics, geophysical flows, neuroscience, and combustion analogs. Future directions include applying the framework to exotic and transitional dynamics (e.g., non-Newtonian turbulence), integrating with control strategies informed by active mechanisms, detecting spurious terms to aid model discovery, and exploring identification of local balances even when global governing equations are incomplete or unknown. Broader adoption should pair the method with careful validation against established theory and experiments.

• Sensitivity to distributional assumptions: GMM presumes Gaussian structure; in the optics case the method did not identify the full Raman time-delay response, suggesting limited sensitivity under non-Gaussian data.
• Dependence on governing equation representation and derivative estimation: The identified balances are tied to the chosen form of the equation and accurate computation of its terms; different representations (e.g., variables) may change interpretability.
• Need for validation: Authors emphasize careful validation to ensure the discovered balances reproduce expected results; the method is intended to augment, not supplant, physical expertise.
• Potential cluster redundancy/ambiguity: Addressed via SPCA, but cluster selection and sparsity thresholds can influence identified regimes, especially in multiscale systems.