Behind general terms like adversarial attacks and generative adversarial networks, such criteria as AI resilience and AI robustness gained a crucial importance, in trying to make the emergent AI tools reliable, energy-efficient and safe. From mathematical perspective, these problems boil down to finding a mathematically sound quantification of the smallest-sufficient amounts of the perturbations of function arguments that lead to the largest possible perturbations of the approximated function values. Here, an entropy-optimising perspective on adversarial algorithms from AI is proposed to attack this problem. It is shown that adopting this perspective helps proving computational conditions for the global optimality and uniqueness of adversarial attacks based on cheaply verifiable mathematical criteria. Further, it is shown how this perspective can be used for developing self-attacking learning algorithms, that generate optimal new data points for training, replacing one of the trained agents with this mathematical criterion. On a broad selection of various synthetic and real-life problems from hydrodynamics and biomedicine, it is shown that such self-attacking learning algorithms allow training orders-of-magnitude simpler and cheaper models with superior performance, and can be used to directly train the optimal controls in complex systems, requiring only a small fraction of the training data and outperforming a set of contemporary AI tools as comprehensive as the author was able to find, including boosted random forests, deep neural networks, and foundational models based on transformer architectures, both in terms of complexity (measured as the model descriptor length) and predictive performance.
Solving partial differential equations is one of the most traditional tasks in scientific computing. In this work, we consider numerical solutions of initial value problems partly or entirely given by linear PDEs and how to compute solutions with a method we refer to as Rational Approximation of Exponential Integration (REXI). REXI replaces a typically sequential timestepping method with a sum of rational terms, leading to the possibility of parallelizing over this sum. Hence, this method can potentially exploit additional degrees of parallelization for scaling problems limited in their spatial scalability to largescale supercomputers. We present the "unified REXI" method in which we show algebraic equivalence to other methods developed up to five decades ago. Such methods cover, e.g., diagonalization of the Butcher table for implicit Runge-Kutta methods, Cauchy-contour integration-based methods, and direct approximations. In the present work, we target the hyperbolic problems considered to be a particularly challenging task. We provide for the first time a deep numerical investigation, discussion, and comparison of all these methods. In particular, we account for numerical problems and, if possible, workarounds for them. Finally, we demonstrate and compare the performance of REXI with off-the-shelf time integrators using the nonlinear shallow-water equations on the rotating sphere on a high-performance computing system. While previous REXI studies have focused on exposing more parallelism to enable faster time to solution, we also consider computing resource efficiency at prescribed accuracy and find that diagonalized lower-order Gauss Runge-Kutta methods (formulated as REXI) are compelling highly efficient methods leading to a 64-fold reduction of the required computational resources compared to existing work.
The Riemann problem is fundamental in the computational modeling of hyperbolic partial differential equations, enabling the development of stable and accurate upwind schemes. While exact solvers provide robust upwinding fluxes, their high computational cost necessitates approximate solvers. Although approximate solvers achieve accuracy in many scenarios, they produce inaccurate solutions in certain cases. To overcome this limitation, we propose constructing neural network-based surrogate models, trained using supervised learning, designed to map interior and exterior conservative state variables to the corresponding exact flux. Specifically, we propose two distinct approaches: one utilizing a vanilla neural network and the other employing a bi-fidelity neural network. The performance of the proposed approaches is demonstrated through applications to one-dimensional and two-dimensional partial differential equations, showcasing their robustness and accuracy.
We design and investigate a variety of multigrid solvers for high-order local discontinuous Galerkin methods applied to elliptic interface and multiphase Stokes problems. Using the template of a standard multigrid V-cycle, we consider a variety of element-wise block smoothers, including Jacobi, multi-coloured Gauss-Seidel, processor-block Gauss-Seidel, and with special interest, smoothers based on sparse approximate inverse (SAI) methods. In particular, we develop SAI methods that: (i) balance the smoothing of velocity and pressure variables in Stokes problems; and (ii) robustly handles high-contrast viscosity coefficients in multiphase problems. Across a broad range of two- and three-dimensional test cases, including Poisson, elliptic interface, steady-state Stokes, and unsteady Stokes problems, we examine a multitude of multigrid smoother and solver combinations. In every case, there is at least one approach that matches the performance of classical geometric multigrid algorithms, e.g., 4 to 8 iterations reduce the residual by 10 orders of magnitude. We also discuss their relative merits with regard to simplicity, robustness, computational cost, and parallelisation.
Our focus is on simulating the dynamics of non-interacting particles including the effects of an external potential, which, under certain assumptions, can be formally described by the Dean-Kawasaki equation. The Dean-Kawasaki equation can be solved numerically using standard finite volume methods. However, the numerical approximation implicitly requires a sufficiently large number of particles to ensure the positivity of the solution and accurate approximation of the stochastic flux. To address this challenge, we extend hybrid algorithms for particle systems to scenarios where the density is low. The aim is to create a hybrid algorithm that switches from a finite volume discretization to a particle-based method when the particle density falls below a certain threshold. We develop criteria for determining this threshold by comparing higher-order statistics obtained from the finite volume method with particle simulations. We then demonstrate the use of the resulting criteria for dynamic adaptation in both two- and three-dimensional spatial settings in the absence of an external potential. Finally we consider the dynamics when an external potential is included.
