Weiwen Wang

Weiwen Wang, Shaoqin Zheng

We propose a fully-decoupled arbitrarily high-order time-stepping scheme for the anisotropic phase-field dendritic crystal growth model. The scheme combines an auxiliary-variable formulation with algebraically stable Runge-Kutta methods and satisfies a discrete energy dissipation law. To address the computational bottleneck arising from the coupled linear system in existing high-order schemes, a matrix diagonalization technique is introduced to transform the coupled linear elliptic system into a set of independent constant-coefficient elliptic equations. The resulting equations can be solved separately and in parallel, thereby improving computational efficiency. Numerical experiments in both two and three dimensions are presented to verify the convergence, energy stability, and efficiency of the proposed scheme. Comparisons with the original coupled formulation demonstrate the effectiveness of the matrix diagonalization strategy, while additional tests illustrate the advantages of high-order temporal discretizations. Simulations under different anisotropy coefficients, latent heat parameters, rotation angles, and initial nucleus shapes are also presented to investigate their effects on dendritic morphology.

Zhao-Rong Lai, Weiwen Wang

Invariant risk minimization (IRM) is an arising approach to generalize invariant features to different environments in machine learning. While most related works focus on new IRM settings or new application scenarios, the mathematical essence of IRM remains to be properly explained. We verify that IRM is essentially a total variation based on $L^2$ norm (TV-$\ell_2$) of the learning risk with respect to the classifier variable. Moreover, we propose a novel IRM framework based on the TV-$\ell_1$ model. It not only expands the classes of functions that can be used as the learning risk and the feature extractor, but also has robust performance in denoising and invariant feature preservation based on the coarea formula. We also illustrate some requirements for IRM-TV-$\ell_1$ to achieve out-of-distribution generalization. Experimental results show that the proposed framework achieves competitive performance in several benchmark machine learning scenarios.