Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) have proven effective in solving partial differential equations (PDEs), especially when some data are available by blending seamlessly data and physics. However, extending PINNs to high-dimensional and even high-order PDEs encounters significant challenges due to the computational cost associated with automatic differentiation in the residual loss. Herein, we address the limitations of PINNs in handling high-dimensional and high-order PDEs by introducing Hutchinson Trace Estimation (HTE). Starting with the second-order high-dimensional PDEs ubiquitous in scientific computing, HTE transforms the calculation of the entire Hessian matrix into a Hessian vector product (HVP). This approach alleviates the computational bottleneck via Taylor-mode automatic differentiation and significantly reduces memory consumption from the Hessian matrix to HVP. We further showcase HTE's convergence to the original PINN loss and its unbiased behavior under specific conditions. Comparisons with Stochastic Dimension Gradient Descent (SDGD) highlight the distinct advantages of HTE, particularly in scenarios with significant variance among dimensions. We further extend HTE to higher-order and higher-dimensional PDEs, specifically addressing the biharmonic equation. By employing tensor-vector products (TVP), HTE efficiently computes the colossal tensor associated with the fourth-order high-dimensional biharmonic equation, saving memory and enabling rapid computation. The effectiveness of HTE is illustrated through experimental setups, demonstrating comparable convergence rates with SDGD under memory and speed constraints. Additionally, HTE proves valuable in accelerating the Gradient-Enhanced PINN (gPINN) version as well as the Biharmonic equation. Overall, HTE opens up a new capability in scientific machine learning for tackling high-order and high-dimensional PDEs.

Federated Learning for Generalization, Robustness, Fairness: A Survey and Benchmark

Federated learning has emerged as a promising paradigm for privacy-preserving collaboration among different parties. Recently, with the popularity of federated learning, an influx of approaches have delivered towards different realistic challenges. In this survey, we provide a systematic overview of the important and recent developments of research on federated learning. Firstly, we introduce the study history and terminology definition of this area. Then, we comprehensively review three basic lines of research: generalization, robustness, and fairness, by introducing their respective background concepts, task settings, and main challenges. We also offer a detailed overview of representative literature on both methods and datasets. We further benchmark the reviewed methods on several well-known datasets. Finally, we point out several open issues in this field and suggest opportunities for further research. We also provide a public website to continuously track developments in this fast advancing field: https://github.com/WenkeHuang/MarsFL.

Generalizable Heterogeneous Federated Cross-Correlation and Instance Similarity Learning

Federated learning is an important privacy-preserving multi-party learning paradigm, involving collaborative learning with others and local updating on private data. Model heterogeneity and catastrophic forgetting are two crucial challenges, which greatly limit the applicability and generalizability. This paper presents a novel FCCL+, federated correlation and similarity learning with non-target distillation, facilitating the both intra-domain discriminability and inter-domain generalization. For heterogeneity issue, we leverage irrelevant unlabeled public data for communication between the heterogeneous participants. We construct cross-correlation matrix and align instance similarity distribution on both logits and feature levels, which effectively overcomes the communication barrier and improves the generalizable ability. For catastrophic forgetting in local updating stage, FCCL+ introduces Federated Non Target Distillation, which retains inter-domain knowledge while avoiding the optimization conflict issue, fulling distilling privileged inter-domain information through depicting posterior classes relation. Considering that there is no standard benchmark for evaluating existing heterogeneous federated learning under the same setting, we present a comprehensive benchmark with extensive representative methods under four domain shift scenarios, supporting both heterogeneous and homogeneous federated settings. Empirical results demonstrate the superiority of our method and the efficiency of modules on various scenarios.

Model-based Deep Reinforcement Learning for Dynamic Portfolio Optimization

Dynamic portfolio optimization is the process of sequentially allocating wealth to a collection of assets in some consecutive trading periods, based on investors' return-risk profile. Automating this process with machine learning remains a challenging problem. Here, we design a deep reinforcement learning (RL) architecture with an autonomous trading agent such that, investment decisions and actions are made periodically, based on a global objective, with autonomy. In particular, without relying on a purely model-free RL agent, we train our trading agent using a novel RL architecture consisting of an infused prediction module (IPM), a generative adversarial data augmentation module (DAM) and a behavior cloning module (BCM). Our model-based approach works with both on-policy or off-policy RL algorithms. We further design the back-testing and execution engine which interact with the RL agent in real time. Using historical {\em real} financial market data, we simulate trading with practical constraints, and demonstrate that our proposed model is robust, profitable and risk-sensitive, as compared to baseline trading strategies and model-free RL agents from prior work.