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In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic)…
Fast and accurate simulation of dynamical systems is a fundamental challenge across scientific and engineering domains. Traditional numerical integrators often face a trade-off between accuracy and computational efficiency, while existing…
Full $d$-manifold DMFT with numerically exact solvers has remained computationally prohibitive for spin-orbit materials due their scaling and severe sign problem, forcing the community to rely on simplified one- and three-band models that…
With rapid progress in deep learning, neural networks have been widely used in scientific research and engineering applications as surrogate models. Despite the great success of neural networks in fitting complex systems, two major…
Understanding strongly correlated systems is essential for advancing quantum chemistry and materials science, yet conventional methods like Density Functional Theory (DFT) often fail to capture their complex electronic behavior. To address…
With the rapid advancement of diffusion-based generative models, Stable Diffusion (SD) has emerged as a state-of-the-art framework for high-fidelity im-age synthesis. However, existing SD models suffer from suboptimal feature aggregation,…
The Density Matrix Renormalization Group (DMRG) has become a powerful numerical method that can be applied to low-dimensional strongly correlated fermionic and bosonic systems. It allows for a very precise calculation of static, dynamic and…
Artificial intelligence has transformed the seismic community with deep learning models (DLMs) that are trained to complete specific tasks within workflows. However, there is still lack of robust evaluation frameworks for evaluating and…
In large-scale excitatory neuronal networks, rapid synchronization manifests as {multiple firing events (MFEs)}, mathematically characterized by a finite-time blow-up of the neuronal firing rate in the mean-field Fokker-Planck equation.…
Recent advances in deep learning has witnessed many innovative frameworks that solve high dimensional mean-field games (MFG) accurately and efficiently. These methods, however, are restricted to solving single-instance MFG and demands…
Numerical solvers for PDEs often struggle to balance computational cost with accuracy, especially in multiscale and time-dependent systems. Neural operators offer a promising way to accelerate simulations, but their practical deployment is…
Many consequential real-world systems, like wind fields and ocean currents, are dynamic and hard to model. Learning their governing dynamics remains a central challenge in scientific machine learning. Dynamic Mode Decomposition (DMD)…
In their preprint Anders et al. [arXiv:1004.0510] propose a crucial modification of the Bosonic Dynamical Mean-Field Theory (B-DMFT) derived by us [Phys. Rev. B 77, 235106 (2008)]. Here we show that the modification consists of two steps…
We present a numerical solution of the dynamical mean field theory of infinite-dimensional equilibrium liquids established in [Phys. Rev. Lett. 116, 015902 (2016)]. For soft sphere interactions, we obtain the numerical solution by an…
This article presents a general framework for recovering missing dynamical systems using available data and machine learning techniques. The proposed framework reformulates the prediction problem as a supervised learning problem to…
The nonequilibrium Green's functions (NEGF) approach is a versatile theoretical tool, which allows to describe the electronic structure, spectroscopy and dynamics of strongly correlated systems. The applicability of this method is, however,…
We present an accurate, efficient and massively parallel finite-element code, DFT-FE, for large-scale ab-initio calculations (reaching $\sim 100,000$ electrons) using Kohn-Sham density functional theory (DFT). DFT-FE is based on a local…
Multi-step forecasting (MSF) in time-series, the ability to make predictions multiple time steps into the future, is fundamental to almost all temporal domains. To make such forecasts, one must assume the recursive complexity of the…
We show how to use diagrammatic techniques to compute the weak-coupling perturbation series of the self-consistent solution to a Dynamical Mean Field Theory (DMFT) problem. This approach constitutes an alternative to using diagrammatic…
A new field of numerical astrophysics is introduced which addresses the solution of large, multidimensional structural or slowly-evolving problems (rotating stars, interacting binaries, thick advective accretion disks, four dimensional…