Source-linked AI summary
Data-driven identification of parametric partial differential equations
Samuel Rudy, Alessandro Alla, Steven L. Brunton, J. Nathan Kutz
TL;DR
The paper addresses the difficulty of identifying PDEs when coefficients vary over time or space and must be separated from the governing dynamics. It introduces group-sparse regression, specifically SGTR, to discover parsimonious parametric PDEs while allowing varying coefficient series. Across the demonstrated examples, SGTR outperformed GLASSO in identifying active PDE terms, with numerical differentiation and library choice remaining important limitations.
Problem
Existing PDE discovery methods were not equipped to identify non-constant coefficients and disambiguate governing evolution from time-dependent parametric dependencies.
Method
The method uses Sequential Grouped Threshold Ridge Regression with groups organized by PDE terms, allowing arbitrary time-series or spatial coefficient variation while enforcing sparse active terms.
Results
SGTR outperformed GLASSO in correctly identifying active PDE terms across the paper’s examples, while GLASSO commonly introduced extra small-coefficient terms.
Takeaways & Limitations
The framework provides a parsimonious approach for separating PDE dynamics from spatial or temporal parametric dependence.
Takeaways & Limitations
Identification remains constrained by numerical differentiation accuracy and by whether the candidate library contains suitable building blocks for the true model.
Abstract
from arXiv · showhide
In this work we present a data-driven method for the discovery of parametric partial differential equations (PDEs), thus allowing one to disambiguate between the underlying evolution equations and their parametric dependencies. Group sparsity is used to ensure parsimonious representations of observed dynamics in the form of a parametric PDE, while also allowing the coefficients to have arbitrary time series, or spatial dependence. This work builds on previous methods for the identification of constant coefficient PDEs, expanding the field to include a new class of equations which until now have eluded machine learning based identification methods. We show that group sequentially thresholded ridge regression outperforms group LASSO in identifying the fewest terms in the PDE along with their parametric dependency. The method is demonstrated on four canonical models with and without the introduction of noise.
1. Introduction.
The paper motivates discovering PDEs whose coefficients vary with time or space, extending constant-coefficient identification to separate governing dynamics from parametric dependence.
- Motivation: Existing automated discovery techniques had primarily addressed PDEs with constant parameters rather than explicitly time-dependent parametrizations.The paper identifies this as a gap in prior PDE discovery methods.
- Motivation: Parametric PDE discovery must distinguish the evolution law from parameters that change over time or vary continuously.The motivating model is u_t = N(u, u_x, u_xx, ..., µ(t)).
- Significance: Characterizing parametric dependence also matters for numerical simulation, future-state prediction, and model reduction in time-independent and time-dependent PDEs.These applications remain relevant even when the governing equations are already known.
- Contribution: The proposed framework uses group sparsity to allow coefficient variation while keeping the active PDE terms consistent.This design targets parsimonious, interpretable representations of observed dynamics.
- Approach: The method extends sparse regression for PDE discovery and combines constant-coefficient identification, group-sparsity regression, and model selection.Model selection balances accuracy against the number of active PDE terms.
2. Methods.
The methods extend sparse PDE discovery to parametric equations whose coefficients vary across time or space while preserving a shared set of active terms. Group sparsity and sequential ridge thresholding support parsimonious identification and model selection.
- Identification of constant coefficient PDEs: PDE-FIND builds an overcomplete library of candidate functions from spatial-time measurements and uses sparse regression to identify constant-coefficient PDEs.The resulting problem is a large overdetermined linear system, Ax = b, whose sparse solution gives the PDE.
- Group sparsity: Group sparsity replaces individual coefficient selection with selection of groups representing coefficient functions across time or space.The method seeks a small number of nonzero groups while allowing arbitrary coefficient values within each group.
- Sequential grouped thresholding: SGTR adapts sequentially thresholded ridge regression to group sparsity because GLASSO performs poorly for identifying PDEs.Groups with sufficiently small relevance are removed iteratively, then the remaining groups are refit to obtain unbiased coefficient estimates.
- Parametric PDE identification: Parametric PDE identification seeks both the nonzero terms and the coefficient values at each timestep or spatial location, with the active terms sharing one sparsity pattern.Time-dependent problems use separate regressions for each timestep, while spatial dependence is handled by grouping according to spatial coordinate.
- Parametric PDE identification: The block-diagonal library matrix groups columns corresponding to the same PDE term, and cross-validation uses 20% of each timestep’s data to select λ.For spatially varying coefficients, the same construction groups by spatial coordinate instead of time.
- Model selection: The AIC-inspired loss balances predictive fit against sparsity, while increasing the threshold or penalty moves models from overfit to parsimonious and finally overly sparse.Figure 2 evaluates models generated from 50 SGTR tolerances and 50 GLASSO parameter values.
3. Computational Results of Parametric PDE Discovery.
The method is evaluated on temporally and spatially parametric PDEs, including Burgers, Navier–Stokes, advection–diffusion, and Kuramoto–Sivashinsky examples with noise. SGTR generally identifies the active terms more reliably than GLASSO, although fourth-order derivatives make noisy Kuramoto–Sivashinsky identification difficult.
- 3.1. Burgers’ Equation with Diffusive Regularization: SGTR correctly identifies the two active terms in parametric Burgers’ equation without noise and with 1% noise, whereas GLASSO adds incorrect terms and fails to recover the PDE dependencies.The Burgers’ coefficient varies sinusoidally in time, while diffusion provides regularization.
- 3.2. Navier-Stokes: Flow around a cylinder: SGTR and GLASSO both correctly identify the active terms in the Navier–Stokes example with a Reynolds-number change from 100 to 75.The parameter changes halfway through the cylinder-flow simulation, and identified coefficient time series are shown for noise-free and 1% noisy data.
- 3.3. Spatially Dependent Advection-Diffusion Equation: Both SGTR and GLASSO correctly identify the active terms of the spatially dependent advection–diffusion equation in noise-free and 1% noisy datasets.Spatial dependence is represented by replacing time-dependent loadings ξ_j(t) with spatial functions ξ_j(x).
- 3.4. Spatially Dependent Kuramoto-Sivashinsky Equation: For spatially dependent Kuramoto–Sivashinsky dynamics, SGTR identifies the correct terms with 0.01% noise but substantial coefficient error, while 1% noise prevents correct active-term identification.The fourth-order derivative makes numerical evaluation with noise highly challenging.
4. Discussion.
The method improves active-term identification over GLASSO, while its extensions to spatial and temporal coefficient variation remain constrained by data partitioning and differentiation accuracy.
- SGTR outperformed GLASSO in correctly identifying active PDE terms, whereas GLASSO often added extra terms with small coefficient values.Post-discovery thresholding is not assumed valid because coefficient magnitude need not reflect term importance after normalization.
- The approach partitions data by timesteps or spatial locations, producing coefficients that vary in time or space.A sufficiently fine grid could potentially support coefficients varying in both space and time, but with reduced resolution.
- Accurate numerical differentiation remains a constraint on identifying dynamics with sparse regression, especially for PDEs requiring derivatives across multiple variables.The discussion identifies automatic differentiation and neural-network-based differentiation as promising ways to improve robustness on noisy data.
- Existing physical-law discovery methods generally study equations of the form u_t = N(u, x, t), leaving parametric dependencies insufficiently addressed.This motivates distinguishing the governing evolution equation from parameters that vary during measurement.