The McKean-Vlasov equation (MVE) describes the collective behavior of particles subject to drift, diffusion, and mean-field interaction. In physical systems, the interaction term can be singular, i.e. it diverges when two particles collide. Notable examples of such interactions include the Coulomb interaction, fundamental in plasma physics, and the Biot-Savart interaction, present in the vorticity formulation of the 2DNavier-Stokes equation (NSE) in fluid dynamics. Solving MVEs that involve singular interaction kernels presents a significant challenge, especially when aiming to provide rigorous theoretical guarantees. In this work, we propose a novel approach based on the concept of entropy dissipation in the underlying system.

The McKean-Vlasov equation (MVE) describes the collective behavior of particles subject to drift, diffusion, and mean-field interaction. In physical systems, the interaction term can be singular, i.e. it diverges when two particles collide. Notable examples of such interactions include the Coulomb interaction, fundamental in plasma physics, and the Biot-Savart interaction, present in the vorticity formulation of the 2D Navier-Stokes equation (NSE) in fluid dynamics. Solving MVEs that involve singular interaction kernels presents a significant challenge, especially when aiming to provide rigorous theoretical guarantees. In this work, we propose a novel approach based on the concept of entropy dissipation in the underlying system.

Real-world data contains aleatoric uncertainty - irreducible noise arising from imperfect measurements or from incomplete knowledge about the data generation process. Mean variance estimation (MVE) networks can learn this type of uncertainty but require ad-hoc regularization strategies to avoid overfitting and are unable to predict epistemic uncertainty (model uncertainty). Conversely, Bayesian neural networks predict epistemic uncertainty but are notoriously difficult to train due to the approximate nature of Bayesian inference. We propose to cooperatively train a variance network with a Bayesian neural network and demonstrate that the resulting model disentangles aleatoric and epistemic uncertainties while improving the mean estimation. We demonstrate the effectiveness and scalability of this method across a diverse range of datasets, including a time-dependent heteroscedastic regression dataset we created where the aleatoric uncertainty is known. The proposed method is straightforward to implement, robust, and adaptable to various model architectures.

The McKean-Vlasov equation (MVE) describes the collective behavior of particles subject to drift, diffusion, and mean-field interaction. In physical systems, the interaction term can be singular, i.e. it diverges when two particles collide. Notable examples of such interactions include the Coulomb interaction, fundamental in plasma physics, and the Biot-Savart interaction, present in the vorticity formulation of the 2D Navier-Stokes equation (NSE) in fluid dynamics. Solving MVEs that involve singular interaction kernels presents a significant challenge, especially when aiming to provide rigorous theoretical guarantees. In this work, we propose a novel approach based on the concept of entropy dissipation in the underlying system.

We extend the concept of self-consistency for the Fokker-Planck equation (FPE) to the more general McKean-Vlasov equation (MVE). While FPE describes the macroscopic behavior of particles under drift and diffusion, MVE accounts for the additional inter-particle interactions, which are often highly singular in physical systems. Two important examples considered in this paper are the MVE with Coulomb interactions and the vorticity formulation of the 2D Navier-Stokes equation. We show that a generalized self-consistency potential controls the KL-divergence between a hypothesis solution to the ground truth, through entropy dissipation. Built on this result, we propose to solve the MVEs by minimizing this potential function, while utilizing the neural networks for function approximation. We validate the empirical performance of our approach by comparing with state-of-the-art NN-based PDE solvers on several example problems.

Standard metrics of performance of classifiers, such as accuracy and sensitivity, do not reveal the trust or confidence in the predicted labels of data. While other metrics such as the computed probability of a label or the signed distance from a hyperplane can act as a trust measure, these are subjected to heuristic thresholds. This paper presents a convex optimization-based supervised classifier that sequentially partitions a dataset into several ellipsoids, where each ellipsoid contains nearly all points of the same label. By stating classification rules based on this partitioning, Bayes' formula is then applied to calculate a trust score to a label assigned to a test datapoint determined from these rules. The proposed Sequential Ellipsoidal Partitioning Classifier (SEP-C) exposes dataset irregularities, such as degree of overlap, without requiring a separate exploratory data analysis. The rules of classification, which are free of hyperparameters, are also not affected by class-imbalance, the underlying data distribution, or number of features. SEP-C does not require the use of non-linear kernels when the dataset is not linearly separable. The performance, and comparison with other methods, of SEP-C is demonstrated on the XOR-problem, circle dataset, and other open-source datasets.

In model-based reinforcement learning, planning with an imperfect model of the environment has the potential to harm learning progress. But even when a model is imperfect, it may still contain information that is useful for planning. In this paper, we investigate the idea of using an imperfect model selectively. The agent should plan in parts of the state space where the model would be helpful but refrain from using the model where it would be harmful. An effective selective planning mechanism requires estimating predictive uncertainty, which arises out of aleatoric uncertainty, parameter uncertainty, and model inadequacy, among other sources. Prior work has focused on parameter uncertainty for selective planning. In this work, we emphasize the importance of model inadequacy. We show that heteroscedastic regression can signal predictive uncertainty arising from model inadequacy that is complementary to that which is detected by methods designed for parameter uncertainty, indicating that considering both parameter uncertainty and model inadequacy may be a more promising direction for effective selective planning than either in isolation.