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 article 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 (FedPINNs) and federated deep operator networks (FedDeepONets). We further introduce various data generation methods to control the degree of nonindependent and identically distributed (non-i.i.d.) 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. In addition, we propose a measure of weight divergence and develop a theoretical framework to establish growth bounds for weight divergence in FL compared with centralized learning. To demonstrate the effectiveness of our methods, we conducted ten experiments, including two on function approximation, five PDE problems on FedPINN, and four 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.

Modeling the evolution of physical dynamics is a foundational problem in science and engineering, and it is regarded as the modeling of an operator mapping between infinite-dimensional functional spaces. Operator learning methods, learning the underlying infinite-dimensional operator in a high-dimensional latent space, have shown significant potential in modeling physical dynamics. However, there remains insufficient research on how to approximate an infinite-dimensional operator using a finite-dimensional parameter space. Inappropriate dimensionality representation of the underlying operator leads to convergence difficulties, decreasing generalization capability, and violating the physical consistency. To address the problem, we present Neural Manifold Operator (NMO) to learn the invariant subspace with the intrinsic dimension to parameterize infinite-dimensional underlying operators. NMO achieves state-of-the-art performance in statistical and physical metrics and gains 23.35% average improvement on three real-world scenarios and four equation-governed scenarios across a wide range of multi-disciplinary fields. Our paradigm has demonstrated universal effectiveness across various model structure implementations, including Multi-Layer Perceptron, Convolutional Neural Networks, and Transformers. Experimentally, we prove that the intrinsic dimension calculated by our paradigm is the optimal dimensional representation of the underlying operators. We release our code at https://github.com/AI4EarthLab/Neural-Manifold-Operators.