9 papers
ATHENA: Agentic Team for Hierarchical Evolutionary Numerical Algorithms
Juan Diego Toscano, Daniel T. Chen, George Em Karniadakis
Progress in computational science depends on complex numerical workflows that must faithfully encode physical laws, yet translating conceptual insight into reliable code remains a…
Explicit Construction of Approximate Kolmogorov Superpositions with C2 Smoothness
Lunji Song, Zilan Cheng, Juan Diego Toscano +1
We explicitly construct an approximate version of the Kolmogorov superpositions, which is composed of C2-inner and outer functions, and can approximate an arbitrary alpha Holder co…
GRAFT-ATHENA: Self-Improving Agentic Teams for Autonomous Discovery and Evolutionary Numerical Algorithms
Juan Diego Toscano, Zhaojie Chai, George Em Karniadakis
Scientific discovery can be modeled as a sequence of probabilistic decisions that map physical problems to numerical solutions. Recent agentic AI systems automate individual scient…
A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learning
Juan Diego Toscano, Daniel T. Chen, Vivek Oommen +2
Residual-based adaptive strategies are widely used in scientific machine learning but remain largely heuristic. We introduce a unifying variational framework that formalizes these…
FMEnets: Flow, Material, and Energy networks for non-ideal plug flow reactor design
Chenxi Wu, Juan Diego Toscano, Khemraj Shukla +7
We propose FMEnets, a physics-informed machine learning framework for the design and analysis of non-ideal plug flow reactors. FMEnets integrates the fundamental governing equation…
KKANs: Kurkova-Kolmogorov-Arnold Networks and Their Learning Dynamics
Juan Diego Toscano, Li-Lian Wang, George Em Karniadakis
Inspired by the Kolmogorov-Arnold representation theorem and Kurkova's principle of using approximate representations, we propose the Kurkova-Kolmogorov-Arnold Network (KKAN), a ne…