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Polymer reference interaction site model (PRISM) theory, a descendent of Ornstein-Zernike liquid state theory, is a powerful tool to predict the structure and thermodynamics of equilibrium polymer systems, but its accuracy and applicability…
We propose a method to compute an approximation of the moments of a discrete-time stochastic polynomial system. We use the Carleman linearization technique to transform this finite-dimensional polynomial system into an infinite-dimensional…
Convergence acceleration of flow simulations to their steady states at lower Mach numbers can be achieved via preconditioning the lattice Boltzmann (LB) schemes that alleviate the associated numerical stiffness, which have so far been…
We present a deep learning approach for computing multi-phase solutions to the semiclassical limit of the Schr\"odinger equation. Traditional methods require deriving a multi-phase ansatz to close the moment system of the Liouville…
The lattice Boltzmann equation (LBE), rooted in kinetic theory, provides a powerful framework for capturing complex flow behaviour by describing the evolution of single-particle distribution functions (PDFs). Despite its success, solving…
The Chemical Master Equation (CME) is used to stochastically model biochemical reaction networks, under the Markovian assumption. The low-order statistical moments induced by the CME are often the key quantities that one is interested in.…
Machine learning (ML) has been well applied to studying equilibrium phase transition models, by accurately predicating critical thresholds and some critical exponents. Difficulty will be raised, however, for integrating ML into…
We propose a well-posed Maxwell-type boundary condition for the linear moment system in half-space. As a reduction of the Boltzmann equation, the moment equations are available to model Knudsen layers near a solid wall, where proper…
The concept of equilibrium is a general tool to fill the gap between macroscopic and mesoscopic information, both within kinetic systems and kinetic schemes. This work explores the use of equilibria to devise numerical boundary conditions…
Machine learning techniques not only offer efficient tools for modelling dynamical systems from data, but can also be employed as frontline investigative instruments for the underlying physics. Nontrivial information about the original…
End-to-end learning has become a widely applicable and studied problem in training predictive ML models to be aware of their impact on downstream decision-making tasks. These end-to-end models often outperform traditional methods that…
In this study, we employ the recently developed recurrence microstate probabilities as features to improve accuracy of several well-established machine learning (ML) algorithms. These algorithms are applied to classify discrete and…
The Boltzmann equation has been a driving force behind significant mathematical research over the years. Its challenging theoretical complexity, combined with a wide variety of current scientific and technological problems that require…
Atmospheric models used for weather and climate prediction are traditionally formulated in a deterministic manner. In other words, given a particular state of the resolved scale variables, the most likely forcing from the sub-grid scale…
Continuous-time neural processes are performant sequential decision-makers that are built by differential equations (DE). However, their expressive power when they are deployed on computers is bottlenecked by numerical DE solvers. This…
The Lattice Boltzmann method (LBM) is a well-established mesoscopic approach for simulating fluid dynamics by evolving particle distribution functions on discrete lattices. While the LBM is highly parallelizable on classical hardware, its…
Many unsteady flows exhibiting complex dynamics are nevertheless characterized by emergent large-scale coherence in space and time. Reduced-order models based on Galerkin projection of the governing equations onto an orthogonal modal basis…
We present a detailed description of the essentially entropic lattice Boltzmann model. The entropic lattice Boltzmann model guarantees unconditional numerical stability by iteratively solving the nonlinear entropy evolution equation. In…
Turbulence closure modeling using machine learning is at an early crossroads. The extraordinary success of machine learning (ML) in a variety of challenging fields has given rise to justifiable optimism regarding similar transformative…
Numerical approaches to the correlated electron problem have achieved considerable success, yet are still constrained by several bottlenecks, including high order polynomial or exponential scaling in system size, long autocorrelation times,…