Federated scientific machine learning for approximating functions and solving differential equations with data heterogeneity

Handi Zhang; Langchen Liu; Lu Lu

Home/Archive/arXiv

arXiv · 2410.13141

Federated scientific machine learning for approximating functions and solving differential equations with data heterogeneity

Handi Zhang·Langchen Liu·Lu Lu

View original source ↗Read source document ↗Export metadata ↓

Abstract

By leveraging neural networks, the emerging field of scientific machine learning (SciML) offers novel approaches to address complex problems governed by partial differential equations (PDEs). In practical applications, challenges arise due to the distributed essence of data, concerns about data privacy, or the impracticality of transferring large volumes of data. Federated learning (FL), a decentralized framework that enables the collaborative training of a global model while preserving data privacy, offers a solution to the challenges posed by isolated data pools and sensitive data issues. Here, this paper explores the integration of FL and SciML to approximate complex functions and solve differential equations. We propose two novel models: federated physics-informed neural networks (FedPINN) and federated deep operator networks (FedDeepONet). We further introduce various data generation methods to control the degree of non-independent and identically distributed (non-iid) data and utilize the 1-Wasserstein distance to quantify data heterogeneity in function approximation and PDE learning. We systematically investigate the relationship between data heterogeneity and federated model performance. Additionally, we propose a measure of weight divergence and develop a theoretical framework to establish growth bounds for weight divergence in federated learning compared to traditional centralized learning. To demonstrate the effectiveness of our methods, we conducted 10 experiments, including 2 on function approximation, 5 PDE problems on FedPINN, and 3 PDE problems on FedDeepONet. These experiments demonstrate that proposed federated methods surpass the models trained only using local data and achieve competitive accuracy of centralized models trained using all data.

Explore related subjects

cs.LG physics.comp-ph

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Handi Zhang, Langchen Liu, Lu Lu. 2024-10-17. Federated scientific machine learning for approximating functions and solving differential equations with data heterogeneity. https://arxiv.org/abs/2410.13141

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rollout Total Correlation for Deep Reinforcement Learning

Learning task-relevant representations is crucial for reinforcement learning. Recent approaches aim to learn such representations by improving the temporal consistency in the observed transitions. However, they only consider individual transitions and can fail to achieve long-term consistency. Instead, we argue that capturing aspects of the state that correlate with other states and actions of the trajectory---even more distant in the future---could further help in extracting task-relevant information. Hence, in this paper we investigate how to learn representations by maximizing the rollout total correlation, the correlation among all learned representations and actions within the trajectories produced by the agent. For improving rollout total correlation, we propose to combine two complementary lower bounds based on a generative and a discriminative model, combined with a simple and effective technique of chunk-wise mini-batching. Furthermore, we propose an intrinsic reward based on the learned representation for better exploration. Experimental evaluations on a set of challenging image-based simulated control tasks show that our method achieves better sample efficiency, and robustness to both white noise and natural video backgrounds compared to leading baselines.

cs.LG↗

arXiv2026

Reward Shaping to Mitigate Reward Hacking in RLHF

Reinforcement learning from human feedback (RLHF) is widely used to align large language models (LLMs) with human preferences. However, RLHF remains vulnerable to \emph{reward hacking}, whereby a policy exploits imperfections in the reward function instead of learning the intended behavior, thereby undermining alignment. Although reward shaping can stabilize RLHF training and partially mitigate reward hacking, shaping methods and their underlying design principles have not been systematically investigated. To address this gap, we conduct a comprehensive study of prevalent reward-shaping techniques. Our analysis identifies two key design principles: (1) the reinforcement-learning reward should be bounded, and (2) it should grow rapidly at first and then gradually saturate. Motivated by these principles, we propose Preference as Reward (PAR), a novel method that uses the latent preferences encoded in the reward model as the reinforcement-learning signal. We further show that PAR possesses two variance-reduction properties that stabilize RLHF training and substantially widen the practical window for early stopping. Our evaluation consists of two parts. First, we compare PAR with several other reward-shaping strategies using Proximal Policy Optimization (PPO) as the reinforcement-learning algorithm and Gemma2-2B as the base model. Second, we compare PAR with the vanilla baseline (i.e., unshaped reward) across four base models and four reinforcement-learning algorithms. In the first set of experiments, PAR consistently outperforms other reward-shaping methods and also reflects high data efficiency and robustness. The second set of experiments shows that PAR is particularly effective for actor-critic RL algorithms when value estimates become unstable and demonstrates its effectiveness across different base models. The code is available at https://github.com/PorUna-byte/PAR.

cs.LG↗

arXiv2026

Trajectory Entropy Reinforcement Learning for Robust Robot Motor Skill Learning

Simplicity is a critical inductive bias for designing data-driven controllers, especially when robustness is important. Despite the impressive results of deep reinforcement learning in complex control tasks, it is prone to capturing intricate and spurious correlations between observations and actions, leading to failure under slight perturbations to the environment. To tackle this problem, in this work we introduce a novel inductive bias towards simple policies in reinforcement learning. The simplicity inductive bias is introduced by minimizing the entropy of entire action trajectories, corresponding to the number of bits required to describe information in action trajectories after the agent observes state trajectories. Our reinforcement learning agent, Trajectory Entropy Reinforcement Learning, is optimized to minimize the trajectory entropy while maximizing rewards. We show that the trajectory entropy can be effectively estimated by learning a variational parameterized action prediction model, and use the prediction model to construct an information-regularized reward function. Furthermore, we construct a practical algorithm that enables the joint optimization of models, including the policy and the prediction model. Experimental evaluations on several high-dimensional locomotion tasks show that our learned policies produce more cyclical and consistent action trajectories, and achieve superior performance, and robustness to noise and dynamic changes than the state-of-the-art.

cs.LG↗