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Robust Discovery of Coarse-Grained Continuum Equations from Microscopic Dynamics
Partha Sarathi Mondal, Manav Kumar Jalan, Anish Kumar, Shradha Mishra
TL;DR
The paper asks how reliably governing PDEs can be discovered from coarse-grained spatiotemporal data under limited data, large function libraries, and noise. It applies PDE-SINDy with stability selection to Allen–Cahn and Glauber Ising data, finding that more data suppresses spurious terms while larger libraries and noise impair robust recovery. For the Ising model, stringent selection recovers a Model-A-like equation reproducing phase-separation and domain-growth features.
Problem
The paper examines how data amount, function-library size, and noise affect discovery of governing PDEs from coarse-grained spatiotemporal data.
Method
The study applies PDE-SINDy with sparse candidate libraries, stability selection, and library bagging to Allen–Cahn and coarse-grained Glauber Ising data.
Results
More data suppresses spurious selections, whereas larger libraries and increased noise weaken separation between governing and competing terms.
Takeaways & Limitations
Stringent selection for the Glauber Ising model recovers a Model-A-like equation that reproduces dynamical and statistical features of phase separation and domain growth.
Abstract
from arXiv · showhide
The discovery of governing partial differential equations (PDEs) directly from spatiotemporal data has emerged as a powerful tool for understanding the dynamics of complex systems. In this work, we apply PDE-SINDy to well-known phase-separating systems and examine how its performance depends on the amount of available data, the size of the function library, and the presence of noise. Our results show that the accuracy of equation discovery depends strongly on the amount of available data. Although the correct equation can be identified with limited data, several spurious terms also acquire finite selection probabilities. As the amount of data increases, these spurious terms are progressively suppressed, leading to a more robust identification of the governing equation. In contrast, increasing the size of the function library adversely affects the efficiency of equation discovery. Further, for the Glauber spin-flip Ising model, we show that the selection probabilities reveal a hierarchy of equations with varying levels of complexity. A sufficiently stringent selection threshold recovers a Model-A-like dynamical equation that accurately reproduces the dynamical and statistical features of phase separation and domain growth.
I. Introduction
Collective phenomena emerge across scales much larger than individual constituents, motivating mesoscopic continuum descriptions and data-driven discovery of governing equations. PDE-SINDy provides an interpretable route to infer coarse-grained dynamics from spatiotemporal data.
- Motivation: Collective phenomena such as phase separation, pattern formation, and flocking arise from microscopic interactions across larger length and time scales.Mesoscopic field theories focus on smoothly varying variables to capture essential dynamics while avoiding irrelevant microscopic information.
- Motivation: Continuum descriptions use a few relevant mesoscopic fields to study kinetic pathways and mechanisms driving collective processes.
- Data-driven discovery: Machine learning enables direct inference of coarse-grained dynamics from data, including symbolic regression methods that recover explicit governing equations.
- Data-driven discovery: Neural networks can learn complex dynamics but are difficult to interpret and typically require large training datasets.
- Study design: The work applies PDE-SINDy with stability selection and library bagging to coarse-grained Glauber Ising data with inherent stochasticity.The Ising model has an established coarse-grained continuum description, providing a test of the framework under noisy data conditions.
II. Methodology
The methodology infers continuum dynamics by representing an unknown function of a field and its spatial derivatives with a sparse candidate library. PDE-SINDy constructs derivative-based features from spatiotemporal field data and estimates coefficients using normalized sparse regression with thresholding.
- Framework: PDE-SINDy infers equations for scalar order-parameter and coarse-grained magnetization fields from spatiotemporal data.The study examines the Allen–Cahn equation and two-dimensional Glauber Ising model.
- Framework: The unknown dynamical function f is assumed to have a sparse representation in a predefined library of candidate functions.The candidate functions depend on the field and its spatial derivatives.
- Library construction: The library is constructed from the field m(r, t) and spatial derivatives, with symmetry and conservation laws available to restrict candidate functions.
- Regression setup: The temporal derivative is computed by forward differences, while candidate functions are evaluated using a 9-point central difference scheme and flattened into library columns.
- Regression setup: Each library column is independently normalized before STRidge iteratively removes coefficients below the cutoff λ through ridge regression and thresholding.This sparse-regression procedure determines the coefficient vector ξ for the discovered equation.
A. Allen-Cahn Equation
The Allen–Cahn equation serves as a benchmark for testing PDE-SINDy with stability selection. The study evaluates how data availability, candidate-library size, and noise affect recovery accuracy.
- Benchmark system: The Allen–Cahn equation, or Model A, benchmarks recovery for systems governed by a single non-conserved order parameter.Its broad applicability makes successful recovery a generic validation of the framework.
- Evaluation factors: The benchmark examines the effects of available data, candidate-library size, and noise on qualitative and quantitative equation recovery.
- Candidate library: The candidate library contains polynomial terms in m through m6, second-order derivative terms, and all possible products, yielding 18 candidate functions.
1. Effect of data size
PDE-SINDy consistently identifies the correct Allen–Cahn terms across data fractions, but coefficient accuracy improves with more data. Reduced data worsens library conditioning and increases multicollinearity, making regression more sensitive to perturbations.
- Qualitative recovery: For all data fractions, the signs of the coefficients for m, m3, and ∇2m are correctly recovered.
- Qualitative recovery: The terms m, m3, and ∇2m are retained with unit selection probability across the tested data fractions.
- Quantitative recovery: Recovered coefficient magnitudes depend strongly on data fraction, with correct equilibrium values obtained only for sufficiently large data fractions and thresholds.The ∇2m coefficient also deviates from unity at smaller data fractions, affecting interfacial properties such as surface tension, interface width, and interface velocity.
- Overall effect: Increasing data size improves quantitative coefficient accuracy while the correct terms remain identifiable across the tested data fractions.
- Regression conditioning: Reducing the data fraction increases both the condition number and VIF, indicating greater multicollinearity among candidate functions.The resulting sensitivity to small numerical perturbations explains larger deviations in recovered coefficients.
2. Effect of Library Size
Expanding the candidate library to 39 terms makes equation recovery less reliable, especially when only a limited fraction of data is available.
- Selection analysis: Selection probabilities are compared for Θ18 and Θ39 at ε = 0.25 and ε = 1.0 using the threshold Πth = 0.80.The three Allen–Cahn terms are highlighted against the remaining candidate terms.
- Library expansion: The candidate library is expanded to 39 terms by adding higher-order spatial derivatives and their combinations with polynomial terms.The resulting library is denoted Θ39 and is compared with the 18-term library Θ18.
- Effect on recovery: The larger candidate library significantly compromises the reliability of the recovered equation when the available data are limited.This effect is considerably more pronounced under reduced-data conditions.
3. Effect of noise
Noise is introduced into the order-parameter field to test equation recovery, and increasing noise degrades both model selection and coefficient accuracy.
- Noise setup: Additive Gaussian white noise is introduced into the order-parameter field, with results reported for amplitudes s = 0.01 and s = 0.02.The noise is delta-correlated, and its strength is scaled by the standard deviation of the order-parameter field.
- Selection probabilities: At s = 0.01, the three Allen–Cahn terms retain unit selection probability, but m5 and m2∇2m also attain high probabilities over a finite threshold range.These additional terms prevent unique identification of the correct governing equation.
- Selection probabilities: For s = 0.02 and ¯λ ≥ 0.40, no candidate term satisfies the selection criterion.The regression algorithm can no longer reliably distinguish the governing dynamics from the added noise.
- Recovered coefficients: At s = 0.01, the Allen–Cahn coefficients have the appropriate signs but differ significantly from their exact values, while spurious terms substantially alter the inferred dynamics.Noise therefore affects both qualitative equation identification and quantitative coefficient accuracy.
B. Inference of the Coarse-Grained Equation
Stability selection with library bagging recovers alternative coarse-grained equations for Glauber spin-flip dynamics, and a stringent threshold selects a Model-A-like equation that reproduces phase separation and domain growth.
- Inference framework: The study combines stability-selection PDE-SINDy with library bagging to infer coarse-grained Glauber dynamics from microscopic simulation data.The candidate library used for regression is denoted Θ14.
- Recovered equations: At Πth = 0.80, Model-I contains only m, m3, and ∇2m, matching the leading-order Model-A equation.The paper contrasts this with a less stringent threshold, Πth = 0.60, yielding Model-II.
- Dynamical validation: Model-I exhibits domain coarsening with curvature-driven interface motion, whereas Model-II does not exhibit phase separation.The comparison is made by simulating both recovered equations from random homogeneous initial conditions.
- Statistical validation: The recovered Model-I dynamics follows the expected domain-growth law l(t) ∼t1/2 for the Glauber Ising model.The characteristic length is obtained from the normalized equal-time two-point correlation function.
IV. Discussion
The paper studies PDE-SINDy with stability selection for phase-separating systems, using simulated Allen–Cahn and coarse-grained Glauber Ising data to assess robustness under data, library, and noise changes.
- Systems and data: The study examines Allen–Cahn and two-dimensional Glauber Ising systems, whose relevant slow variables are scalar order parameters or local coarse-grained magnetization fields.The governing continuum equations are inferred from spatiotemporal data using PDE-SINDy.
- Systems and data: Glauber Ising trajectories are generated by MCMC and converted into local magnetization fields using block coarse-graining.The dataset contains 10 distinct trajectories with 800 snapshots per trajectory after an initial 50 MCS and spacing of 4 MCS.
- Stability-selection procedure: Different data fractions are formed by randomly selecting εNtot consecutive snapshots, which are used to compute derivatives and construct the candidate library matrix.The resulting regression problem is ∂tm = Θξ.
- Stability-selection procedure: Stability selection repeatedly subsamples 50% of regression rows and performs sparse regression 1000 times for each threshold λ.Each candidate’s selection probability measures its robustness to variations in sampled data.
- Stability-selection procedure: For Glauber dynamics, library bagging randomly retains 70% of candidate functions before stability selection to improve robustness.For the Θ14 library, 150 independent library-bagging realizations are generated.
C. Stability selection results for different data fractions
With the 18-term library, the known Allen–Cahn terms remain stable across data fractions, while limited data also gives spurious terms appreciable selection probabilities.
- Data-fraction effects: The Allen–Cahn terms m, m3, and ∇2m retain unit selection probability across a broad threshold range for every data fraction ε.These are the three terms constituting the Allen–Cahn equation.
- Data-fraction effects: Several spurious terms attain high selection probabilities when only limited data are available.Thus, the correct equation may remain identifiable while competing terms are not fully suppressed.
D. Stability selection results for different library size
Larger candidate libraries make relevant and irrelevant terms harder to distinguish, while noise weakens both true-term stability and overall equation-discovery robustness.
- Library-size effects: Enlarging the candidate library substantially increases the number of competing terms with high selection probability.The comparison considers 18- and 39-term libraries at ε = 0.25 and ε = 1.00.
- Library-size effects: At ε = 0.25, the larger library produces additional terms with selection probabilities comparable to true terms over a broad threshold range.This increases the probability of incorrect model selection.
- Library-size effects: At ε = 1.00, most competing terms from the larger library are effectively suppressed.The distinction between relevant and irrelevant terms therefore becomes clearer with complete data.
- Noise effects: At noise strength s = 0.01, m, m3, and ∇2m remain stable broadly, but m5 and m(∇m)2 become weakly stable over limited threshold ranges.These higher-order terms illustrate noise-associated spurious selection.
- Noise effects: Increasing noise to s = 0.02 substantially decreases candidate selection probabilities and narrows the threshold range supporting the true terms.The paper identifies this as a reduction in the overall robustness of equation discovery.