We for the first time leverage deep learning approaches to solve forward and inverse problems of two-dimensional laminar flows for power-law fluids. We propose a deep-learning framework, called Power-Law-Fluid-Net (PL-Net). We develop a surrogate model to solve the forward problems of the power-law fluids, and solve the inverse problems utilizing only a small set of measurement data under the assumption that boundary conditions (BCs) can be partially known. In the design of the methods, we incorporate the hard boundary condition constraints to accelerate the iteration of stochastic gradient descent methods for minimizing loss functions. For the forward problems, by incorporating the constitutive parameters into the input variables of neural networks, the PL-Net serves as a surrogate model for simulating the pressure-driven flows inside pipes having cross sections of varying shapes. We investigate the influences of the BC type, activation function type, and number of collocation points on the accuracy of numerical solutions. For the inverse problems, the PL-Net infers the physical quantities or constitutive parameters from a small number of measurements of flow field variables. The BCs of the inverse problems can even be partially known. We demonstrate the effects of BC type, number of sensors, and noise level on accuracy of inferred quantities. Computational examples indicate the high accuracy of the PL-Net in tackling both the forward and inverse problems of the power-law fluids.

We give an outline of the content of our recent monograph “The Shock Development Problem” which analyzes the development of shocks in fluids. The framework is that of Euler’s equations of the mechanics of compressible fluids. We first set up the mathematical problem as an initial-boundary value problem for a nonlinear hyperbolic system of partial differential equations with a free boundary and singular initial conditions. Then we describe the main mathematical methods which we introduce to solve the problem. Some of these methods are geometric and have to do with a geometric structure defined on spacetime by the fluid and its interaction with the background spacetime structure. Also, with the nature of the free boundary and the jump conditions there which include a non-linear jump condition. The central method however is analytic and arises from the need to handle singular integrals appearing in the energy identities. Ancient solutions play an important role in studying singularities. These are special solutions to an evolution equation that exists for all time −∞ < t ≤ T, with T ≤ +∞. They typically appear as blow-up limits near a singularity. We will discuss some of the recent developments regarding the classification of ancient solutions to geometric flows, in particular the mean curvature flow and the Ricci flow. For many years, the use of the Wiener-Hopf technique in acoustics and other physical problems was the only manifestation of an application of the Riemann- Hilbert formalism. However, in the last 50 years, this formalism and its natural generalization called the ∂-bar formalism have appeared in a large number of problems in mathematics and mathematical physics. In this paper, the impact of these formalisms in three separate areas is reviewed: first, the development of a novel, hybrid numerical-analytical method for solving boundary value problems for linear and integrable nonlinear PDEs, known as the unified transform or the Fokas method (www.wikipedia.org/wiki/Fokas_method). Second, the introduction of a new algorithm in nuclear medical imaging called the attenuated spline reconstruction technique (aSRT). Third, a novel approach to the Lindelöf hypothesis, which is a close relative of the Riemann hypothesis. We provide a concise overview of the genesis and impact of Fourier analysis in mathematics. We review some important results that have driven research during the last 50 years, and we discuss recent advances in multilinear aspects of the theory. In this paper, we consider the basic version of Reinforcement Learning (RL) that involves computing optimal data driven (adaptive) policies for Markovian decision process with unknown transition probabilities. We provide a brief survey of the state of the art of the area, and we compare the performance of the classic UCB policy of Burnetas and Katehakis [10] with a new policy developed herein that we call MDPDeterministic Minimum Empirical Divergence (MDP-DMED) and a method based on Posterior sampling (MDP-PS). The prediction of various phenomena through models and computation is a necessity in numerous real-life problems. The computation of singular phenomena (interfaces, shocks, defects, cracks, etc.) arises in many complex systems and poses many challenges. For computing such phenomena, it is natural to seek methods that are able to detect them and to devote the necessary computational recourses to their accurate resolution. These phenomena are very interesting for their importance and applicability, but also for the challenge they pose in mathematical research. From a computational perspective, we need to design algorithms that are fast but also reliable. In this paper, we consider examples of such phenomena modelled by nonlinear partial differential equations and discrete microscopic systems. Since the often weak solutions of PDEs related to these problems are not unique, we demonstrate that “natural ad hoc” computational methods might predict irrelevant solutions. Since numerical methods perturb the mathematical model, mathematical analysis emerges as a necessary tool which ensures that our computational methods approximate physically relevant solutions. We discuss in more detail three nonlinear problems: (i) a problem related to design and analysis of approximate atomistic-continuum energies to atomistic models arising in crystalline materials; (ii) a problem of cell interaction within a fibrin medium. Motivated by experiments, we demonstrate that the combination of sophisticated mathematical modelling and numerical analysis leads to reliable computational predictions and reveals the real mechanisms of the observed interactions; and (iii) a problem arising in statistical inference of solutions to nonlinear hyperbolic systems. To compute measure-valued solutions, we propose new discrete kinetic models, and we study corresponding kinetic formulations of viscous and inviscid conservation laws. We survey various results that are related to generalizations of dyadic maximal operators. The Bellman functions that are associated with the dyadic maximal operators in Euclidean spaces comprise a very useful tool in the study of these operators. These notions may be generalized in the context of tree-like families of measurable subsets in a non-atomic probability space. Certain sharp inequalities for these Bellman functions are presented, whose proof uses an effective linearization method developed by the author. Analogous estimates can be obtained for the Bellman functions that are associated with local inequalities as well as for Lorentz-type Bellman functions. Finally, certain stability results for extremal or approximate extremal functions are presented; such properties of extremal functions are needed for a deeper analysis of Bellman function estimates. In this paper, we consider imaging problems that can be cast in the form of an underdetermined linear...

Although stochastic fractional partial differential equations have received increasing attention in the last decade, the parameter estimation of these equations has been seldom reported in literature. In this paper, we propose a pseudo-likelihood approach to estimating the parameters of stochastic time-fractional diffusion equations, whose forward solver has been investigated very recently by Gunzburger, Li, and Wang (2019). Our approach can accurately recover the fractional order, diffusion coefficient, as well as noise magnitude given the discrete observation data corresponding to only one realization of driving noise. When only partial data is available, our approach can also attain acceptable results for intermediate sparsity of observation.

Convolutional neural operator is a CNN-based architecture recently proposed to enforce structure-preserving continuous-discrete equivalence and enable the genuine, alias-free learning of solution operators of PDEs. This neural operator was demonstrated to outperform for certain cases some baseline models such as DeepONet, Fourier neural operator, and Galerkin transformer in terms of surrogate accuracy. The convolutional neural operator, however, seems not to be validated for few-shot learning. We extend the model to few-shot learning scenarios by first pre-training a convolutional neural operator using a source dataset and then adjusting the parameters of the trained neural operator using only a small target dataset. We investigate three strategies for adjusting the parameters of a trained neural operator, including fine-tuning, low-rank adaption, and neuron linear transformation, and find that the neuron linear transformation strategy enjoys the highest surrogate accuracy in solving PDEs such as Kuramoto-Sivashinsky equation, Brusselator diffusion-reaction system, and Navier-Stokes equations.