Zhuo Chen

Zhuo Chen, Jacob McCarran, Esteban Vizcaino et al.

Partial differential equations (PDEs) are instrumental for modeling dynamical systems in science and engineering. The advent of neural networks has initiated a significant shift in tackling these complexities though challenges in accuracy persist, especially for initial value problems. In this paper, we introduce the $\textit{Time-Evolving Natural Gradient (TENG)}$, generalizing time-dependent variational principles and optimization-based time integration, leveraging natural gradient optimization to obtain high accuracy in neural-network-based PDE solutions. Our comprehensive development includes algorithms like TENG-Euler and its high-order variants, such as TENG-Heun, tailored for enhanced precision and efficiency. TENG's effectiveness is further validated through its performance, surpassing current leading methods and achieving $\textit{machine precision}$ in step-by-step optimizations across a spectrum of PDEs, including the heat equation, Allen-Cahn equation, and Burgers' equation.

Youneng Bao, Yiping Liu, Zhuo Chen et al.

Driven by the ``scale-is-everything'' paradigm, modern machine learning increasingly demands ever-larger datasets and models, yielding prohibitive computational and storage requirements. Dataset distillation mitigates this by compressing an original dataset into a small set of synthetic samples, while preserving its full utility. Yet, existing methods either maximize performance under fixed storage budgets or pursue suitable synthetic data representations for redundancy removal, without jointly optimizing both objectives. In this work, we propose a joint rate-utility optimization method for dataset distillation. We parameterize synthetic samples as optimizable latent codes decoded by extremely lightweight networks. We estimate the Shannon entropy of quantized latents as the rate measure and plug any existing distillation loss as the utility measure, trading them off via a Lagrange multiplier. To enable fair, cross-method comparisons, we introduce bits per class (bpc), a precise storage metric that accounts for sample, label, and decoder parameter costs. On CIFAR-10, CIFAR-100, and ImageNet-128, our method achieves up to $170\times$ greater compression than standard distillation at comparable accuracy. Across diverse bpc budgets, distillation losses, and backbone architectures, our approach consistently establishes better rate-utility trade-offs.

Jiangran Wang, Zhuo Chen, Di Luo et al.

We develop a spacetime neural network method with second order optimization for solving quantum dynamics from the high dimensional Schrödinger equation. In contrast to the standard iterative first order optimization and the time-dependent variational principle, our approach utilizes the implicit mid-point method and generates the solution for all spatial and temporal values simultaneously after optimization. We demonstrate the method in the Schrödinger equation with a self-normalized autoregressive spacetime neural network construction. Future explorations for solving different high dimensional differential equations are discussed.