<div>Federated learning (FL) is a decentralized privacy-preserving learning technique in which clients learn a joint collaborative model through a central aggregator without sharing their data. In this setting, all clients learn a single common predictor (FedAvg), which does not generalize well on each client's local data due to the statistical data heterogeneity among clients. In this paper, we address this problem with PersFL, a discrete two-stage personalized learning algorithm. In the first stage, PersFL finds the optimal teacher model of each client during the FL training phase. In the second stage, PersFL distills the useful knowledge from optimal teachers into each user's local model. The teacher model provides each client with some rich, high-level representation that a client can easily adapt to its local model, which overcomes the statistical heterogeneity present at different clients. We evaluate PersFL on CIFAR-10 and MNIST datasets using three data-splitting strategies to control the diversity between clients' data distributions.</div><div><br></div><div>We empirically show that PersFL outperforms FedAvg and three state-of-the-art personalization methods, pFedMe, Per-FedAvg and FedPer on majority data-splits with minimal communication cost. Further, we study the performance of PersFL on different distillation objectives, how this performance is affected by the equitable notion of fairness among clients, and the number of required communication rounds. We also build an evaluation framework with the following modules: Data Generator, Federated Model Generation, and Evaluation Metrics. We introduce new metrics for the domain of personalized FL, and split these metrics into two perspectives: Performance, and Fairness. We analyze the performance of all the personalized algorithms by applying these metrics to answer the following questions: Which personalization algorithm performs the best in terms of accuracy across all the users?, and Which personalization algorithm is the fairest amongst all of them? Finally, we make the code for this work available at https://tinyurl.com/1hp9ywfa for public use and validation.</div>

<div>In this work, we introduce a framework for constructing light-weight neural network approximations to the solution operators for partial differential equations (PDEs). Using a data-driven offline training procedure, the resulting operator network models are able to effectively reduce the computational demands of traditional numerical methods into a single forward-pass of a neural network. Importantly, the network models can be calibrated to specific distributions of input data in order to reflect properties of real-world data encountered in practice. The networks thus provide specialized solvers tailored to specific use-cases, and while being more restrictive in scope when compared to more generally-applicable numerical methods (e.g. procedures valid for entire function spaces), the operator networks are capable of producing approximations significantly faster as a result of their specialization.</div><div><br></div><div>In addition, the network architectures are designed to place pointwise posterior distributions over the observed solutions; this setup facilitates simultaneous training and uncertainty quantification for the network solutions, allowing the models to provide pointwise uncertainties along with their predictions. An analysis of the predictive uncertainties is presented with experimental evidence establishing the validity of the uncertainty quantification schema for a collection of linear and nonlinear PDE systems. The reliability of the uncertainty estimates is also validated in the context of both in-distribution and out-of-distribution test data.</div><div><br></div><div>The proposed neural network training procedure is assessed using a novel convolutional encoder-decoder model, ConvPDE-UQ, in addition to an existing fully-connected approach, DeepONet. The convolutional framework is shown to provide accurate approximations to PDE solutions on varying domains, but is restricted by assumptions of uniform observation data and homogeneous boundary conditions. The fully-connected DeepONet framework provides a method for handling unstructured observation data and is also shown to provide accurate approximations for PDE systems with inhomogeneous boundary conditions; however, the resulting networks are constrained to a fixed domain due to the unstructured nature of the observation data which they accommodate. These two approaches thus provide complementary frameworks for constructing PDE-based operator networks which facilitate the real-time approximation of solutions to PDE systems for a broad range of target applications.</div>