Thus, one important question one can ask here is how welldFnn(ˆµn,ˆνm)approximatesdFnn(µ,ν).

Regression with non-Euclidean responses -- e.g., probability distributions, networks, symmetric positive-definite matrices, and compositions -- has become increasingly important in modern applications. In this paper, we propose deep Fréchet neural networks (DFNNs), an end-to-end deep learning framework for predicting non-Euclidean responses -- which are considered as random objects in a metric space -- from Euclidean predictors. Our method leverages the representation-learning power of deep neural networks (DNNs) to the task of approximating conditional Fréchet means of the response given the predictors, the metric-space analogue of conditional expectations, by minimizing a Fréchet risk. The framework is highly flexible, accommodating diverse metrics and high-dimensional predictors. We establish a universal approximation theorem for DFNNs, advancing the state-of-the-art of neural network approximation theory to general metric-space-valued responses without making model assumptions or relying on local smoothing. Empirical studies on synthetic distributional and network-valued responses, as well as a real-world application to predicting employment occupational compositions, demonstrate that DFNNs consistently outperform existing methods.

We introduce Lagrangian Flow Networks (LFlows) for modeling fluid densities and velocities continuously in space and time. By construction, the proposed LFlows satisfy the continuity equation, a PDE describing mass conservation in its differentiable form. Our model is based on the insight that solutions to the continuity equation can be expressed as time-dependent density transformations via differentiable and invertible maps. This follows from classical theory of the existence and uniqueness of Lagrangian flows for smooth vector fields. Hence, we model fluid densities by transforming a base density with parameterized diffeomorphisms conditioned on time. The key benefit compared to methods relying on numerical ODE solvers or PINNs is that the analytic expression of the velocity is always consistent with changes in density. Furthermore, we require neither expensive numerical solvers, nor additional penalties to enforce the PDE. LFlows show higher predictive accuracy in density modeling tasks compared to competing models in 2D and 3D, while being computationally efficient. As a real-world application, we model bird migration based on sparse weather radar measurements.

FINs can be trained to emulate one-or-more weights that are initialized to approximate closed-form statistical features, and may then be integrated within a larger, more complex features. In this work, we perform the first-ever evaluation of FINs network architecture that obtains the power of the feature, without for biomedical image processing tasks. We begin by training a the strict limitations that would result from including the feature set of FINs to imitate six common radiomics features, and then as an input to the model directly. That is, as part of network compare the performance of networks with and without the FINs fine-tuning, the representation captured by the FIN evolves from the for three experimental tasks: COVID-19 detection from CT scans, static feature representation it was first trained to emulate into an brain tumor classification from MRI scans, and brain-tumor segmentation instantiation that is most effective for the task at hand; For instance, from MRI scans; we find that FINs provide best-in-class a FIN that is designed to emulate Shannon's entropy, may evolve performance for all three tasks, while converging faster and more into a Tsalis entropy representation during fine-tuning.