Extension to broader dissipative PDE classes

Extend the CARE-SAV variational framework to broader classes of dissipative partial differential equations while retaining its structure-preserving properties.

Background

The paper develops and analyzes CARE-SAV for a class of gradient-flow models, including Allen–Cahn, Cahn–Hilliard, phase-field crystal, molecular beam epitaxy, and grain-growth problems. Its theoretical results rely on assumptions such as a self-adjoint nonnegative linear energy operator, a dissipative mobility operator, and a scalar auxiliary-variable reformulation.

The conclusion explicitly leaves open the extension of the same variational perspective to broader classes of dissipative PDEs. This problem concerns determining whether the CARE representation and energy-stable SAV discretization can be generalized beyond the model structures treated in the paper.

References

On the other hand, several questions remain open beyond the present study. In particular, the construction of the CARE space may be further adapted to the evolving solution, allowing the spatial representation to respond dynamically to changes in solution structure. Extending the same variational perspective to broader classes of dissipative PDEs also provides a natural direction for future investigation.

— CARE-SAV: A Conditioning-Aware Random-Feature Framework for Energy-Stable Simulation of Gradient Flows  (2608.25300 - Hu et al., 26 Aug 2026) in Section 7, Discussions and Conclusions