CARE-SAV: A Conditioning-Aware Random-Feature Framework for Energy-Stable Simulation of Gradient Flows

Bingcheng Hu; Zhaoxiang Li

Source-linked AI summary

CARE-SAV: A Conditioning-Aware Random-Feature Framework for Energy-Stable Simulation of Gradient Flows

Bingcheng Hu, Zhaoxiang Li

arXiv:2608.25300v1Aug 26, 2026math.NA

TL;DR

Gradient-flow discretizations must preserve energy dissipation, but conventional prescribed spatial spaces limit representation flexibility. CARE-SAV constructs a compact feature-generated Galerkin space and evolves the dynamics within it, retaining discrete energy stability; the authors identify adaptive spaces and broader dissipative PDEs as future directions.

Problem
Conventional gradient-flow schemes use spatial approximation spaces prescribed in advance, motivating a more flexible representation that retains structure-preserving properties.
Method
CARE-SAV constructs a compact spatial approximation space from flexible candidate features and combines it with a scalar auxiliary variable formulation.
Results
CARE-SAV retains the discrete energy-dissipation structure while providing a flexible alternative to predetermined meshes or bases.
Takeaways & Limitations
Feature representations and rigorous structure-preserving discretization can coexist within a nontraditional spatial representation for dissipative PDE solvers.
Takeaways & Limitations
The study leaves open dynamically adapting the CARE space to evolving solutions and extending the variational perspective to broader dissipative PDE classes.

Abstract

from arXiv · show

Gradient-flow models are characterized by an intrinsic energy-dissipation structure, and faithfully preserving this structure at the discrete level is important for stable and reliable long-time simulation. To this end, we develop a Conditioning-Aware Representation Enhancement with Scalar Auxiliary Variable (CARE-SAV) framework, which constructs a compact spatial approximation space from flexible candidate features and evolves the gradient-flow dynamics directly within this space. The resulting fully discrete scheme preserves the discrete energy-dissipation law while providing a flexible alternative to conventional prescribed spatial discretizations. Rigorous analysis establishes the approximation capability, solvability, stability and convergence of the proposed method. Numerical experiments on representative gradient-flow problems demonstrate its accuracy, robustness and computational efficiency. We believe that CARE-SAV could provide a simple, flexible, and computationally efficient paradigm for structure-preserving discretization of gradient-flow problems.

1. Introduction

Gradient flows dissipate free energy toward equilibrium, motivating discretizations that preserve this structure while allowing more flexible spatial representations than prescribed meshes or bases. CARE-SAV addresses this challenge by constructing a compact feature-generated Galerkin space and retaining structure-preserving properties.

Gradient flows evolve toward decreasing free energy, whose decay supports stable and physically consistent long-time simulation.
Prescribed finite difference, finite element, and spectral spaces limit spatial representations to predetermined approximation classes.
CARE-SAV constructs a compact spatial approximation space from redundant flexible candidate functions while retaining the energy-dissipation structure.
The framework integrates flexible spatial approximation with structure-preserving discretization at the formulation level.
The work establishes solvability, stability, conservation, and convergence properties and evaluates accuracy, robustness, and efficiency through representative experiments.

2. Gradient-Flow Formulation and the SAV Framework

The paper formulates gradient flows and the SAV reformulation, then develops the Crank-Nicolson-based SAV discretization and its discrete energy law. The framework assumes suitable operators and nonlinear-energy positivity, yielding a linear, unconditionally energy-stable temporal scheme while motivating conditioning-aware spatial construction.

2. Gradient-Flow Formulation and the SAV Framework: The formulation considers gradient flows on spatial domains with periodic or homogeneous Neumann boundaries, using the L2 inner product and norm.
2.1. Gradient-Flow Formulation.: The free energy is decomposed into a quadratic component involving a self-adjoint nonnegative operator and a nonlinear contribution.
2.1. Gradient-Flow Formulation.: A nonpositive mobility operator produces monotone free-energy decay for the unforced system, making the energy a Lyapunov functional.
2.1. Gradient-Flow Formulation.: The SAV reformulation requires E1(ϕ)+C0 > 0 and replaces the nonlinear energy contribution with a scalar variable while preserving continuous equivalence.
2.2. The SAV-CN Scheme and Discrete Energy Law.: Crank-Nicolson discretization targets second-order temporal accuracy while retaining a discrete energy-dissipation structure.
2.2. The SAV-CN Scheme and Discrete Energy Law.: Explicit treatment of the nonlinear coefficient yields a linear scheme requiring only linear-system solves at each time step.
2.2. The SAV-CN Scheme and Discrete Energy Law.: The SAV-CN scheme satisfies a discrete energy law, so its modified energy is nonincreasing for any time-step size.
2.2. The SAV-CN Scheme and Discrete Energy Law.: Direct random-feature representations can be ill-conditioned because of correlated or redundant basis functions, motivating conditioning-aware compact spatial construction.

3. Conditioning-Aware Representation Enhancement

CARE constructs a compact, boundary-compatible approximation space from an overcomplete random-feature dictionary by removing numerically redundant directions. A CPQR-based reduction produces a discrete-L2-orthonormal basis for Galerkin discretization.

CARE begins with an overcomplete collection of boundary-compatible Gaussian candidate features for spatial approximation.
Candidate features are sampled through centers and widths, with periodization used to satisfy periodic boundary conditions.
Strongly correlated or redundant candidate directions can make the feature matrices severely ill-conditioned, motivating numerical rank reduction.
CPQR orders features by incremental contribution to the numerical column space, retaining dominant independent directions and discarding nearly dependent ones.
The effective dimension K is selected adaptively from R's diagonal using τQR, with larger τQR producing more aggressive reduction.
The retained CARE basis is orthonormal under the discrete L2(Ω) inner product, giving the reduced Galerkin system an identity mass matrix.

4. The CARE-SAV Method

The CARE-SAV method projects gradient-flow dynamics onto the reduced CARE space and applies SAV with Crank-Nicolson time integration. Explicit nonlinear extrapolation yields a linear, efficiently solvable scheme that retains the SAV energy structure.

CARE-SAV combines CARE Galerkin approximation with the SAV framework to form a reduced spatially semidiscrete system.
The initial condition is obtained by discrete L2 projection onto the CARE space, with coefficients computed using the orthonormal basis.
The CARE Galerkin projection retains the SAV energy-dissipation structure at the spatially semidiscrete level.
Crank-Nicolson discretization with explicit extrapolation defines the fully discrete coefficient-level scheme.
Because the projected nonlinear vector is explicit, the fully discrete system is linear in the unknowns at the next time level.
The fully discrete scheme retains the SAV energy structure, with unique solvability and unconditional energy stability established theoretically.
Each time step requires two linear solves with the same K × K matrix, whose factorization can be reused because it is time independent.

5. Stability, Error and Conditioning Analysis

The analysis establishes solvability, energy stability, conditioning properties, approximation estimates and convergence for CARE-SAV. The method is unconditionally energy stable, second-order accurate in time for fixed CARE spaces, and supports mass conservation under stated conditions.

Theoretical results cover unique solvability, unconditional energy stability, fully discrete convergence and mass conservation for the Cahn-Hilliard case.
Under Assumption 1, the fully discrete CARE-SAV scheme has a unique solution for every ∆t > 0.
The scheme is unconditionally energy stable with respect to the quadrature-based modified SAV energy.
The preserved dissipative quantity is the modified SAV energy, which generally does not exactly equal the discrete physical energy.
For conserved flows, discrete mass conservation follows when the constant function is represented exactly and the mobility operator is assembled structure-preservingly.
CARE improves numerical robustness by replacing redundant candidate directions with a conditioning-aware reduced representation.
CARE truncation separates the original random-feature approximation error from the compression error introduced by retaining leading directions.
The method is second-order accurate in time for a fixed CARE space, while spatial convergence depends on CARE-basis approximation and quadrature consistency.

6. Numerical Experiments

Numerical experiments across five representative gradient-flow models assess CARE-SAV’s accuracy, energy stability, conservation-related behavior, and parameter sensitivity. The reported results show accurate approximations, monotone energy decay where applicable, and stable accuracy as feature richness and QR tolerances vary.

Experiments cover Allen-Cahn, Cahn-Hilliard, phase-field crystal, molecular beam epitaxy, and grain growth models.The study evaluates numerical accuracy and energy stability across five representative gradient-flow problems.
Allen-Cahn Equation: For Allen-Cahn, CARE-SAV closely matches the reference solution at ε = 10^-5, despite thinner interfaces and sharper spatial gradients becoming more challenging as ε decreases.The accompanying accuracy tables report accuracy and computational cost for different ε values.
Allen-Cahn Equation: Allen-Cahn physical and shifted modified SAV energies decrease monotonically and are nearly indistinguishable throughout the simulation.This indicates consistency between the auxiliary variable and the original nonlinear energy.
Cahn-Hilliard Equation: For Cahn-Hilliard, CARE-SAV closely agrees with the reference solution at t = T for ε = 0.1, while both physical and modified SAV energies decrease monotonically.The two Cahn-Hilliard energy curves are visually indistinguishable, indicating strong auxiliary-variable consistency.
Phase-Field Crystal Model: For PFC, CARE-SAV accurately captures spatial structures at t = T for r = -0.5, and both physical and modified SAV energies decrease monotonically for different r values.The energy curves remain nearly indistinguishable, showing that the auxiliary variable closely tracks the original nonlinear energy.
Molecular Beam Epitaxy Model: The MBE manufactured-solution experiment confirms accurate resolution, but its forcing term removes the unforced gradient-flow setting, so physical energy need not be monotone.This example primarily assesses numerical accuracy rather than energy dissipation.
Grain Growth Model: In grain growth, active grains decrease from 16 to 13 while physical and modified SAV energies decrease monotonically and remain visually indistinguishable.The reported dynamics include migrating and smoothing boundaries, shrinking small grains, and expanding larger grains.
Parameter Sensitivity: Increasing candidate features improves accuracy with diminishing returns, while tested rank-revealing QR tolerances preserve stable accuracy after redundant directions are removed.These sensitivity results support feature reduction without substantial accuracy loss.

7. Discussions and Conclusions

The paper argues that flexible feature representations can coexist with structure-preserving discretization when the generated feature space is treated as a Galerkin space. It identifies adaptive spatial representations and extensions to reduced or data-driven dissipative-PDE descriptions as future directions.

Discussion and conclusions: Treating the generated feature space as a genuine Galerkin space retains gradient-flow dissipation within a nontraditional spatial representation.This separates structure preservation from dependence on a predetermined mesh or basis.
Discussion and conclusions: The framework suggests that representation flexibility and rigorous numerical structure can coexist in adaptable structure-preserving solvers for dissipative PDEs.
Future directions: Future work could adapt the CARE space to evolving solution structure so that the spatial representation responds dynamically to changes in the solution.
Future directions: Extending the variational perspective to broader dissipative PDE classes remains an open direction.
Future directions: The feature-space perspective may support reduced or data-driven descriptions of dissipative-PDE dynamics beyond direct time integration.A related goal is retaining structural information in finite-dimensional dynamical representations.

Appendix A. Proof of Proposition 5.1

The proof establishes nonsingularity of the reduced system matrix by combining dissipativity with nonnegativity, then derives the auxiliary-variable relation needed for Proposition 5.1.

Nonsingularity proof: Taking the inner product with LKx and using dissipativity yields 2(LKx)T GK(LKx) ≤ 0.
Nonsingularity proof: Because the left-hand side is nonnegative, LKx = 0, and the defining equation then implies x = 0.
Nonsingularity proof: Therefore AK is nonsingular.
Auxiliary relation: The equation for qn+1/2 is equivalent to qn+1/2 = GKz, providing the relation used in the proposition proof.
Conclusion: The preceding steps complete the proof of Proposition 5.1.

Appendix B. Proof of Proposition 5.2

The proof uses discrete CARE-basis orthonormality and dissipativity properties to establish the stated algebraic identities and inequalities under the numerical basis assumption.

Proof identities: Discrete orthonormality of the CARE basis, ΨT WΨ = IK, gives the first identity directly.
Numerical basis assumption: If δQ := ∥eΨT W eΨ − IK∥2 < 1, every eigenvalue of eΨT W eΨ lies in [1 − δQ, 1 + δQ].
Dissipativity estimate: The final inequality follows from dissipativity of GK and positive semidefiniteness of LK.

Appendix C. Proof of Lemma 5.1

The appendices establish approximation and error-control ingredients for CARE spaces, including almost-sure density of sampled features, treatment of SAV consistency defects, and a bound combining approximation with quadrature and assembly errors.

CARE approximation: The best nodal approximation from the selected CARE functions is obtained by orthogonal projection onto range(QK).
Feature-space density: Full support on a separable compact parameter space implies that the sampled parameter sequence is almost surely dense.
Feature-space density: For nonperiodic domains, sampled Gaussian radial-basis features are almost surely dense in C(Ω), and therefore in L2(Ω).
Periodic density: For periodic rectangular cells, periodized Gaussian translates have nonzero Fourier coefficients at every frequency and generate a dense span in C(Td).
SAV error analysis: The physical SAV variable is used as the scalar reference, while the projected-energy quantity is introduced only for comparison.
SAV error analysis: The physical reference has a nonzero spatial consistency defect, so no zero-SAV-defect assumption is imposed.
Consistency and stability: The error analysis accounts for vector defects, quadrature and assembly consistency defects, trajectory bounds, local Lipschitz continuity, and stability estimates.
Error bound: The combined defect estimate satisfies sup 0≤t≤T (∥ξG(t)∥ + ∥ξL,U(t)∥ + |ξr(t)|) ≤ C [AM,N,K(T) + QM,N,K(T)].

Appendix F. Proof of Lemma 5.3

The appendix combines a proof of the CARE-SAV error estimate with sensitivity experiments on candidate-feature counts and QR tolerances. The experiments show improved accuracy with larger feature sets but diminishing returns, while accuracy remains stable across tested QR tolerances.

Proof of Lemma 5.3: The proof concludes the stated recurrence using bounds, one-step initialization, and the discrete Gronwall inequality.The argument also permits setting E−1 = E0 for initialization.
Experimental Configuration: The experiments use double-precision Python implementations with a common CARE basis-construction procedure and SAV-CN time discretization.Default configurations are summarized using M, RBF width ranges, τQR, effective dimension K, Δt, and Nt.
Number of Candidate Features: Increasing candidate features generally improves Allen-Cahn and Cahn-Hilliard accuracy, with diminishing returns after a sufficiently rich representation is reached.The tested values are M = 160, 320, 480, 640 and 960.
QR Tolerance Sensitivity: Relative L2 errors remain within the same order of magnitude as τQR varies from 10^-8 to 10^-14.Allen-Cahn errors generally decrease with smaller τQR, whereas Cahn-Hilliard errors vary mildly nonmonotonically.

Appendix I. Data Generator

The data generator constructs reference solutions for periodic benchmark problems using Fourier pseudospectral spatial discretization and ETDRK4 time integration. Refined snapshots are interpolated to CARE collocation points for error measurement.

Reference Solver: Reference solutions use Fourier pseudospectral discretization in space and fourth-order ETDRK4 integration in time.Each scalar equation or vector component is written in semilinear form with the linear operator treated exactly and nonlinear terms evaluated pseudospectrally.
ETDRK4 Update: ETDRK4 advances Fourier coefficients through intermediate states and nonlinear evaluations before forming the next-step solution.The update uses E, Q, f1, f2, and f3 together with nonlinear terms at the intermediate states.
Spectral Evaluation: Fourier-space products are computed mode by mode, nonlinearities are evaluated in physical space, and the two-thirds rule removes aliasing errors.Coefficient functions near the origin use analytic limits or contour averaging to avoid cancellation.
Snapshot Generation: The Fourier grid and reference time step are refined until recorded snapshots are unchanged at the reported accuracy.The resulting fields are saved at prescribed output times and Fourier-interpolated to CARE collocation points.
Availability and Conflicts: The generated and analyzed datasets are available from the corresponding author on request, and the authors report no conflict of interest.