Despite classical statistical theory predicting severe overfitting, modern massively overparameterized neural networks still generalize well. This unexpected property is attributed to the network's so-called implicit bias, which describes its propensity to converge to solutions that generalize effectively, among the many possible that correctly label the training data. The aim of our research is to explore this bias from a new perspective, focusing on how non-linear activation functions contribute to shaping it. First, we introduce a reparameterization which removes a continuous weight rescaling symmetry. Second, in the kernel regime, we leverage this reparameterization to generalize recent findings that relate shallow Neural Networks to the Radon transform, deriving an explicit formula for the implicit bias induced by a broad class of activation functions. Specifically, by utilizing the connection between the Radon transform and the Fourier transform, we interpret the kernel regime's inductive bias as minimizing a spectral seminorm that penalizes high-frequency components, in a manner dependent on the activation function. Finally, in the adaptive regime, we demonstrate the existence of local dynamical attractors that facilitate the formation of clusters of hyperplanes where the input to a neuron's activation function is zero, yielding alignment between many neurons' response functions. We confirm these theoretical results with simulations. All together, our work provides a deeper understanding of the mechanisms underlying the generalization capabilities of overparameterized neural networks and its relation with the implicit bias, offering potential pathways for designing more efficient and robust models.

Solving fluid dynamics equations often requires the use of closure relations that account for missing microphysics. For example, when solving equations related to fluid dynamics for systems with a large Reynolds number, sub-grid effects become important and a turbulence closure is required, and in systems with a large Knudsen number, kinetic effects become important and a kinetic closure is required. By adding an equation governing the growth and transport of the quantity requiring the closure relation, it becomes possible to capture microphysics through the introduction of ``hidden variables'' that are non-local in space and time. The behavior of the ``hidden variables'' in response to the fluid conditions can be learned from a higher fidelity or ab-initio model that contains all the microphysics. In our study, a partial differential equation simulator that is end-to-end differentiable is used to train judiciously placed neural networks against ground-truth simulations. We show that this method enables an Euler equation based approach to reproduce non-linear, large Knudsen number plasma physics that can otherwise only be modeled using Boltzmann-like equation simulators such as Vlasov or Particle-In-Cell modeling.

Plasma supports collective modes and particle-wave interactions that leads to complex behavior in inertial fusion energy applications. While plasma can sometimes be modeled as a charged fluid, a kinetic description is useful towards the study of nonlinear effects in the higher dimensional momentum-position phase-space that describes the full complexity of plasma dynamics. We create a differentiable solver for the plasma kinetics 3D partial-differential-equation and introduce a domain-specific objective function. Using this framework, we perform gradient-based optimization of neural networks that provide forcing function parameters to the differentiable solver given a set of initial conditions. We apply this to an inertial-fusion relevant configuration and find that the optimization process exploits a novel physical effect that has previously remained undiscovered.