Cory Hauck

Hayden Schaeffer, Stanley Osher, Russel Caflisch et al.

We investigate the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis. The restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation. By reducing or compressing the information needed to represent the solution at every step, only the essential dynamics are represented. In many cases, there are natural bases derived from the differential equations which promote sparsity. We find that our method successfully reduces the dynamics of convection equations, diffusion equations, weak shocks, and vorticity equations with high frequency source terms.

Giacomo Dimarco, Cory Hauck, Raphaël Loubère

In this paper we genealize the fast semi-Lagrangian scheme developed in [J. Comput. Phys., Vol. 255, 2013, pp 680-698] to the case of high order reconstructions of the distribution function. The original first order accurate semi-Lagrangian scheme is supplemented with polynomial reconstructions of the distribution function and of the collisional operator leading to an effective high order accurate numerical scheme for all regimes, from extremely rarefied gas to highly collisional siuation. The main idea relies on updating at each time step the extreme points of the distribution function for each velocity of the lattice instead of updating the solution in the cell centers, these extremes points being located at different positions for any fixed velocity of the lattice. The result is a class of scheme which permits to preserve the structure of the solution over very long times compared to existing schemes from the literature. We propose a proof of concept of this new approach along with numerical tests and comparisons with classical numerical methods.

José A. Morales Escalante, Irene M. Gamba, Eirik Endeve et al.

The work presented in this paper is related to the development of positivity preserving Discontinuous Galerkin (DG) methods for Boltzmann - Poisson (BP) computational models of electronic transport in semiconductors. We pose the Boltzmann Equation for electron transport in curvilinear coordinates for the momentum. We consider the 1D diode problem with azimuthal symmetry, which is a 3D plus time problem. We choose for this problem the spherical coordinate system $\vec{p}(|\vec{p}|,μ=cosθ,φ)$, slightly different to the choice in previous DG solvers for BP, because its DG formulation gives simpler integrals involving just piecewise polynomial functions for both transport and collision terms. Applying the strategy of Zhang \& Shu, \cite{ZhangShu1}, \cite{ZhangShu2}, Cheng, Gamba, Proft, \cite{CGP}, and Endeve et al. \cite{EECHXM-JCP}, we treat the collision operator as a source term, and find convex combinations of the transport and collision terms which guarantee the positivity of the cell average of our numerical probability density function at the next time step. The positivity of the numerical solution to the pdf in the whole domain is guaranteed by applying the limiters in \cite{ZhangShu1}, \cite{ZhangShu2} that preserve the cell average but modify the slope of the piecewise linear solutions in order to make the function non - negative. In addition of the proofs of positivity preservation in the DG scheme, we prove the stability of the semi-discrete DG scheme under an entropy norm, using the dissipative properties of our collisional operator given by its entropy inequalities. The entropy inequality we use depends on an exponential of the Hamiltonian rather than the Maxwellian associated just to the kinetic energy.

Ahmed Almeldein, Mohammed Alnaggar, Rick Archibald et al.

The AI for Nuclear Energy workshop at Oak Ridge National Laboratory evaluated the potential of Large Language Models (LLMs) to accelerate fusion and fission research. Fourteen interdisciplinary teams explored diverse nuclear science challenges using ChatGPT, Gemini, Claude, and other AI models over a single day. Applications ranged from developing foundation models for fusion reactor control to automating Monte Carlo simulations, predicting material degradation, and designing experimental programs for advanced reactors. Teams employed structured workflows combining prompt engineering, deep research capabilities, and iterative refinement to generate hypotheses, prototype code, and research strategies. Key findings demonstrate that LLMs excel at early-stage exploration, literature synthesis, and workflow design, successfully identifying research gaps and generating plausible experimental frameworks. However, significant limitations emerged, including difficulties with novel materials designs, advanced code generation for modeling and simulation, and domain-specific details requiring expert validation. The successful outcomes resulted from expert-driven prompt engineering and treating AI as a complementary tool rather than a replacement for physics-based methods. The workshop validated AI's potential to accelerate nuclear energy research through rapid iteration and cross-disciplinary synthesis while highlighting the need for curated nuclear-specific datasets, workflow automation, and specialized model development. These results provide a roadmap for integrating AI tools into nuclear science workflows, potentially reducing development cycles for safer, more efficient nuclear energy systems while maintaining rigorous scientific standards.

Shih-Hsin Wang, Justin Baker, Cory Hauck et al.

The pioneering work of Oono and Suzuki [ICLR, 2020] and Cai and Wang [arXiv:2006.13318] initializes the analysis of the smoothness of graph convolutional network (GCN) features. Their results reveal an intricate empirical correlation between node classification accuracy and the ratio of smooth to non-smooth feature components. However, the optimal ratio that favors node classification is unknown, and the non-smooth features of deep GCN with ReLU or leaky ReLU activation function diminish. In this paper, we propose a new strategy to let GCN learn node features with a desired smoothness -- adapting to data and tasks -- to enhance node classification. Our approach has three key steps: (1) We establish a geometric relationship between the input and output of ReLU or leaky ReLU. (2) Building on our geometric insights, we augment the message-passing process of graph convolutional layers (GCLs) with a learnable term to modulate the smoothness of node features with computational efficiency. (3) We investigate the achievable ratio between smooth and non-smooth feature components for GCNs with the augmented message-passing scheme. Our extensive numerical results show that the augmented message-passing schemes significantly improve node classification for GCN and some related models.