Partial Differential Equation Solver Research Articles

TetraFEM: Numerical Solution of Partial Differential Equations Using Tensor Train Finite Element Method

In this paper, we present a methodology for the numerical solving of partial differential equations in 2D geometries with piecewise smooth boundaries via finite element method (FEM) using a Quantized Tensor Train (QTT) format. During the calculations, all the operators and data are assembled and represented in a compressed tensor format. We introduce an efficient assembly procedure of FEM matrices in the QTT format for curvilinear domains. The features of our approach include efficiency in terms of memory consumption and potential expansion to quantum computers. We demonstrate the correctness and advantages of the method by solving a number of problems, including nonlinear incompressible Navier–Stokes flow, in differently shaped domains.

Simulating Nonlinear Radiation Diffusion Through Quantum Computing

The recent success of the fusion ignition experiment at Lawrence-Livermore National Laboratory has renewed excitement for inertial confinement fusion. Designing such experiments relies on computational modeling using radiation hydrodynamic codes run on massively parallel supercomputers which simulate hydrodynamic flows, radiation diffusion, and thermonuclear burn. Constructing a quantum algorithm for radiation hydrodynamics that provides a quantum speedup is of great interest. A recent quantum algorithm that solves the Navier-Stokes equations of hydrodynamics with a quadratic speedup marks a first step towards this goal. Here we take the next step and present a quantum algorithm for nonlinear radiation diffusion that also has a quadratic speedup. To verify the algorithm we consider radiation striking a cold, optically thick target that generates a Marshak wave in the target. Results of numerical simulation of the quantum algorithm are compared with those of a standard partial differential equation solver and excellent agreement is found.

Parallel Computing Techniques for Solving Large-Scale Partial Differential Equation

Two-layered partial differential equations (PDEs) will be structured in this investigation using a variety of parallel mathematical techniques, such as Red Dull Gauss Seidel and Gauss Seidel. For the solution of this study, the two-layered PDEs of the elliptic and illustrated sorts were selected. Parallel Virtual Machine (PVM) is used on the correspondence among all chips of the Parallel Handling System. PVM is well-known for its item system, which enables several diverse computers to be used as flexible and easy-to-use concurrent handling resources. An evaluation of the parallel exhibition assessment including the Gauss Seidel and Red Gauss Seidel techniques in terms of execution time, speed, capability, ampleness, and transient execution will follow the graphical presentation of the mathematical results. Since the goal of this work was to create a successful Two-Dimensional PDE Solver (TDPDES), performance evaluations are simple. Numerous design and mathematical sectors will see an improvement in their examination and investigation systems thanks to this new, effective TDPDES technology..

Enhancing subsurface multiphase flow simulation with Fourier neural operator

Accurate numerical modeling of multiphase flow in subsurface oil and gas reservoirs is critical for optimizing hydrocarbon recovery. However, traditional physics-based algorithms face substantial computational hurdles due to the need for fine grid resolution and the inherent geological heterogeneity. To overcome these challenges, data-driven surrogate models solving the flow governing partial differential equations (PDEs) offer a promising alternative to enhance the efficiency of hydrocarbon production operations. In this study, we employ the Fourier Neural Operator (FNO) to extract spectral information from the reservoir properties, thereby facilitating the solution of coupled porous flow PDEs. Our focus is on two-phase flow dynamics, specifically exploring how water injection enhances reservoir pressure and displaces oil. This scenario involves solving a set of nonlinearly coupled PDEs with highly heterogeneous coefficients. Numerical results demonstrate that the developed FNO accurately predicts the reservoir pressure distributions. We further observe that the FNO's zero-shot super-resolution capability is sensitive to abrupt local changes in the reservoir pressure near injection and production wells. To enhance its accuracy, we propose a multi-fidelity FNO model that exhibits better adaptability across various grid configurations. After moderate training on graphics processing units (GPUs), the FNO achieves a speedup of three orders of magnitude compared to traditional numerical PDE solvers. Our experiments confirm the FNO's potential to replace repetitive physics-based simulations, significantly advancing computational efficiency in the uncertainty quantification of reservoir performance.

Generalising quantum imaginary time evolution to solve linear partial differential equations

The quantum imaginary time evolution (QITE) methodology was developed to overcome a critical issue as regards non-unitarity in the implementation of imaginary time evolution on a quantum computer. QITE has since been used to approximate ground states of various physical systems. In this paper, we demonstrate a practical application of QITE as a quantum numerical solver for linear partial differential equations. Our algorithm takes inspiration from QITE in that the quantum state follows the same normalised trajectory in both algorithms. However, it is our QITE methodology’s ability to track the scale of the state vector over time that allows our algorithm to solve differential equations. We demonstrate our methodology with numerical simulations and use it to solve the heat equation in one and two dimensions using six and ten qubits, respectively.

Traffic assignment optimization to improve urban air quality with the unified finite-volume physics-informed neural network

Emissions by road transportation constitute the major contributor of air pollutants to threat the public health, especially on mega cities with high-rise buildings and growing population. A practicable and economical remedy is assigning the traffic volume judiciously, termed traffic assignment optimization (TAO). Incorporating the computational fluid dynamics (CFD), this method provides precisely optimized traffic volumes under versatile meteorological conditions. However, the high-dimensional degree of freedoms (DoF) involved in CFD hampers its further application toward the large-scale problem where a massive urban area is considered. With the more complicated urban traffic network, the larger decision variable dimension also impedes the time-efficient acquisition of the optimization results. To alleviate this curse-of-dimensionality, the current work proposes an optimization framework with the surrogate model based on the unified finite-volume physics-informed neural networks (UFV-PINN). The UFV-PINN plays the dual role as the partial differential equation (PDE) solver and the surrogate model for pollutant concentration prediction. It reduces the DoF within the PDE solution process and enables the gradient-based optimization. The current work optimizes the average CO concentration and accumulative travel time in Kowloon peninsula of Hong Kong, using the multiple-gradient descent algorithm (MGDA). This optimization framework is tested with various working conditions. Results indicate that the proposed method attain high accuracy solutions, comparable to the heuristic algorithms. This study is the first attempt to incorporate UFV-PINN into the TAO with air quality consideration. Endowed with the differentiability by UFV-PINN, the framework will facilitate the TAO to provide precise suggestions toward large-scale problems.

A meshless stochastic method for Poisson-Nernst-Planck equations.

A plethora of biological, physical, and chemical phenomena involve transport of charged particles (ions). Its continuum-scale description relies on the Poisson-Nernst-Planck (PNP) system, which encapsulates the conservation of mass and charge. The numerical solution of these coupled partial differential equations is challenging and suffers from both the curse of dimensionality and difficulty in efficiently parallelizing. We present a novel particle-based framework to solve the full PNP system by simulating a drift-diffusion process with time- and space-varying drift. We leverage Green's functions, kernel-independent fast multipole methods, and kernel density estimation to solve the PNP system in a meshless manner, capable of handling discontinuous initial states. The method is embarrassingly parallel, and the computational cost scales linearly with the number of particles and dimension. We use a series of numerical experiments to demonstrate both the method's convergence with respect to the number of particles and computational cost vis-à-vis a traditional partial differential equation solver.

A Label-Free Hybrid Iterative Numerical Transferable Solver for Partial Differential Equations

A Label-Free Hybrid Iterative Numerical Transferable Solver for Partial Differential Equations

Parametric encoding with attention and convolution mitigate spectral bias of neural partial differential equation solvers

Parametric encoding with attention and convolution mitigate spectral bias of neural partial differential equation solvers

Inf-sup neural networks for high-dimensional elliptic PDE problems

Solving high dimensional partial differential equations (PDEs) has historically posed a considerable challenge when utilizing conventional numerical methods, such as those involving domain meshes. Recent advancements in the field have seen the emergence of neural PDE solvers, leveraging deep networks to effectively tackle high dimensional PDE problems. This study introduces Inf-SupNet, a model-based unsupervised learning approach designed to acquire solutions for a specific category of elliptic PDEs. The fundamental concept behind Inf-SupNet involves incorporating the inf-sup formulation of the underlying PDE into the loss function. The analysis reveals that the global solution error can be bounded by the sum of three distinct errors: the numerical integration error, the duality gap of the loss function (training error), and the neural network approximation error for functions within Sobolev spaces. To validate the efficacy of the proposed method, numerical experiments conducted in high dimensions demonstrate its stability and accuracy across various boundary conditions, as well as for both semi-linear and nonlinear PDEs.

Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes

In recent years, there has been widespread adoption of machine learning-based approaches to automate the solving of partial differential equations (PDEs). Among these approaches, Gaussian processes (GPs) and kernel methods have garnered considerable interest due to their flexibility, robust theoretical guarantees, and close ties to traditional methods. They can transform the solving of general nonlinear PDEs into solving quadratic optimization problems with nonlinear, PDE-induced constraints. However, the complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel, and its partial derivatives, a result of the PDE constraint and for which fast algorithms are scarce. The primary goal of this paper is to provide a near-linear complexity algorithm for working with such kernel matrices. We present a sparse Cholesky factorization algorithm for these matrices based on the near-sparsity of the Cholesky factor under a novel ordering of pointwise and derivative measurements. The near-sparsity is rigorously justified by directly connecting the factor to GP regression and exponential decay of basis functions in numerical homogenization. We then employ the Vecchia approximation of GPs, which is optimal in the Kullback-Leibler divergence, to compute the approximate factor. This enables us to compute ϵ \epsilon -approximate inverse Cholesky factors of the kernel matrices with complexity O ( N log d ⁡ ( N / ϵ ) ) O(N\log ^d(N/\epsilon )) in space and O ( N log 2 d ⁡ ( N / ϵ ) ) O(N\log ^{2d}(N/\epsilon )) in time. We integrate sparse Cholesky factorizations into optimization algorithms to obtain fast solvers of the nonlinear PDE. We numerically illustrate our algorithm’s near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Ampère equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs and kernel methods.

Koopman neural operator as a mesh-free solver of non-linear partial differential equations

The lacking of analytic solutions of diverse partial differential equations (PDEs) gives birth to a series of computational techniques for numerical solutions. Although numerous latest advances are accomplished in developing neural operators, a kind of neural-network-based PDE solver, these solvers become less accurate and explainable while learning long-term behaviors of non-linear PDE families. In this paper, we propose the Koopman neural operator (KNO), a new neural operator, to overcome these challenges. With the same objective of learning an infinite-dimensional mapping between Banach spaces that serves as the solution operator of the target PDE family, our approach differs from existing models by formulating a non-linear dynamic system of equation solution. By approximating the Koopman operator, an infinite-dimensional operator governing all possible observations of the dynamic system, to act on the flow mapping of the dynamic system, we can equivalently learn the solution of a non-linear PDE family by solving simple linear prediction problems. We validate the KNO in mesh-independent, long-term, and zero-shot predictions on five representative PDEs (e.g., the Navier-Stokes equation and the Rayleigh-Bénard convection) and three real dynamic systems (e.g., global water vapor patterns and western boundary currents). In these experiments, the KNO exhibits notable advantages compared with previous state-of-the-art models, suggesting the potential of the KNO in supporting diverse science and engineering applications (e.g., PDE solving, turbulence modeling, and precipitation forecasting).

Lagrangian relaxation for continuous‐time optimal control of coupled hydrothermal power systems including storage capacity and a cascade of hydropower systems with time delays

AbstractThis work considers a short‐term, continuous time setting characterized by a coupled power supply system controlled exclusively by a single provider and comprising a cascade of hydropower systems (dams), fossil fuel power stations, and a storage capacity modeled by a single large battery. Cascaded hydropower generators introduce time‐delay effects in the state dynamics, which are modeled with differential equations, making it impossible to use classical dynamic programming. We address this issue by introducing a novel Lagrangian relaxation technique over continuous‐time constraints, constructing a nearly optimal policy efficiently. This approach yields a convex, nonsmooth optimization dual problem to recover the optimal Lagrangian multipliers, which is numerically solved using a limited memory bundle method. At each step of the dual optimization, we need to solve an optimization subproblem. Given the current values of the Lagrangian multipliers, the time delays are no longer active, and we can solve a corresponding nonlinear Hamilton–Jacobi–Bellman (HJB) Partial Differential Equation (PDE) for the optimization subproblem. The HJB PDE solver provides both the current value of the dual function and its subgradient, and is trivially parallelizable over the state space for each time step. To handle the infinite‐dimensional nature of the Lagrange multipliers, we design an adaptive refinement strategy to control the duality gap. Furthermore, we use a penalization technique for the constructed admissible primal solution to smooth the controls while achieving a sufficiently small duality gap. Numerical results based on the Uruguayan power system demonstrate the efficiency of the proposed mathematical models and numerical approach.

Derivation and analysis of lattice Boltzmann form of the mild slope equation

The present study investigates the performance of the Lattice Boltzmann Method (LBM) as an alternative numerical method for simulating wave propagation in coastal zones. While formulations like the Mild-Slope Equation (MSE) remain efficient for coastal engineering applications due to their simplicity, LBM offers an explicit numerical solver for partial differential equations (PDEs). LBM utilizes local operations, making it well-suited for Graphics Processing Unit (GPU) acceleration and consequently efficient simulations. This study examines LBM's performance in several benchmark cases, including wave propagation behind a single breakwater, harbor resonance, refraction of long waves over a parabolic shoal, and wave propagation over an elliptic shoal. The Chapman-Enskog multi-scale expansion was used within a popular 2D LBM scheme to derive the relaxation time and equilibrium distribution function. The results showed a good agreement between LBM predictions and benchmark data. Furthermore, LBM demonstrated efficiency in reducing the computation time, achieving a notable 30–50% decrease in CPU time and an 85% reduction when utilizing a GPU.

Solving forward and inverse problems of contact mechanics using physics-informed neural networks

This paper explores the ability of physics-informed neural networks (PINNs) to solve forward and inverse problems of contact mechanics for small deformation elasticity. We deploy PINNs in a mixed-variable formulation enhanced by output transformation to enforce Dirichlet and Neumann boundary conditions as hard constraints. Inequality constraints of contact problems, namely Karush–Kuhn–Tucker (KKT) type conditions, are enforced as soft constraints by incorporating them into the loss function during network training. To formulate the loss function contribution of KKT constraints, existing approaches applied to elastoplasticity problems are investigated and we explore a nonlinear complementarity problem (NCP) function, namely Fischer–Burmeister, which possesses advantageous characteristics in terms of optimization. Based on the Hertzian contact problem, we show that PINNs can serve as pure partial differential equation (PDE) solver, as data-enhanced forward model, as inverse solver for parameter identification, and as fast-to-evaluate surrogate model. Furthermore, we demonstrate the importance of choosing proper hyperparameters, e.g. loss weights, and a combination of Adam and L-BFGS-B optimizers aiming for better results in terms of accuracy and training time.

Across Time and Space: Senju ’s Approach for Scaling Iterative Stencil Loop Accelerators on Single and Multiple FPGAs

Stencil-based applications play an essential role in high-performance systems as they occur in numerous computational areas, such as partial differential equation solving. In this context, Iterative Stencil Loops (ISLs) represent a prominent and well-known algorithmic class within the stencil domain. Specifically, ISL-based calculations iteratively apply the same stencil to a multi-dimensional point grid multiple times or until convergence. However, due to their iterative and intensive nature, ISLs are highly performance-hungry, demanding specialized solutions. Here, Field Programmable Gate Arrays (FPGAs) represent a valid architectural choice as they enable the design of custom, parallel, and scalable ISL accelerators. Besides, the regular structure of ISLs makes them an ideal candidate for automatic optimization and generation flows. For these reasons, this article introduces Senju , an automation framework for the design of highly parallel ISL accelerators targeting single-/multi-FPGA systems. Given an input description, Senju automates the entire design process and provides accurate performance estimations. The experimental evaluation shows remarkable and scalable results, outperforming single- and multi-FPGA literature approaches under different metrics. Finally, we present a new analysis of temporal and spatial parallelism trade-offs in a real-case scenario and discuss our performance through a single- and novel specialized multi-FPGA formulation of the Roofline Model.

LordNet: An efficient neural network for learning to solve parametric partial differential equations without simulated data

Neural operators, as a powerful approximation to the non-linear operators between infinite-dimensional function spaces, have proved to be promising in accelerating the solution of partial differential equations (PDE). However, it requires a large amount of simulated data, which can be costly to collect. This can be avoided by learning physics from the physics-constrained loss, which we refer to it as mean squared residual (MSR) loss constructed by the discretized PDE. We investigate the physical information in the MSR loss, which we called long-range entanglements, and identify the challenge that the neural network requires the capacity to model the long-range entanglements in the spatial domain of the PDE, whose patterns vary in different PDEs. To tackle the challenge, we propose LordNet, a tunable and efficient neural network for modeling various entanglements. Inspired by the traditional solvers, LordNet models the long-range entanglements with a series of matrix multiplications, which can be seen as the low-rank approximation to the general fully-connected layers and extracts the dominant pattern with reduced computational cost. The experiments on solving Poisson’s equation and (2D and 3D) Navier–Stokes equation demonstrate that the long-range entanglements from the MSR loss can be well modeled by the LordNet, yielding better accuracy and generalization ability than other neural networks. The results show that the Lordnet can be 40× faster than traditional PDE solvers. In addition, LordNet outperforms other modern neural network architectures in accuracy and efficiency with the smallest parameter size.

ANN-based model integrated in thermal-hydraulics codes: A case study of two-phase wall friction model

Accurate prediction of two-phase parameters is essential for the development, operation and safety of nuclear power plants. In this paper, the ANN-based model has been developed, implemented with PDE (Partial Differential Equation) solver in case study of two-phase frictional pressure drop prediction.

An Application-Driven Method for Assembling Numerical Schemes for the Solution of Complex Multiphysics Problems

Within recent years, considerable progress has been made regarding high-performance solvers for partial differential equations (PDEs), yielding potential gains in efficiency compared to industry standard tools. However, the latter largely remains the status quo for scientists and engineers focusing on applying simulation tools to specific problems in practice. We attribute this growing technical gap to the increasing complexity and knowledge required to pick and assemble state-of-the-art methods. Thus, with this work, we initiate an effort to build a common taxonomy for the most popular grid-based approximation schemes to draw comparisons regarding accuracy and computational efficiency. We then build upon this foundation and introduce a method to systematically guide an application expert through classifying a given PDE problem setting and identifying a suitable numerical scheme. Great care is taken to ensure that making a choice this way is unambiguous, i.e., the goal is to obtain a clear and reproducible recommendation. Our method not only helps to identify and assemble suitable schemes but enables the unique combination of multiple methods on a per-field basis. We demonstrate this process and its effectiveness using different model problems, each comparing the resulting numerical scheme from our method with the next best choice. For both the Allen–Cahn and advection equations, we show that substantial computational gains can be attained for the recommended numerical methods regarding accuracy and efficiency. Lastly, we outline how one can systematically analyze and classify a coupled multiphysics problem of considerable complexity with six different unknown quantities, yielding an efficient, mixed discretization that in configuration compares well to high-performance implementations from the literature.

Utilizing optimal physics-informed neural networks for dynamical analysis of nanocomposite one-variable edge plates

This study presents an innovative Physics-Informed Neural Network (PINN) approach designed to predict the dynamic responses of the One-Variable Edge Plate (OVEP), a unique plate structure characterized by an edge defined by arbitrary mathematical functions. The OVEP is constructed from a nanocomposite material reinforced with graphene nanoplatelets. Utilizing an optimized PINN pipeline, this research successfully predicts the vibration characteristics of the OVEP, encompassing linear, nonlinear vibrations, and beating phenomena. The study also demonstrates that the optimal PINN outperforms conventional neural network in terms of stability, accuracy (achieving above 99% accuracy), and efficiency in predicting long-duration vibrations. Additionally, the computational time required for generating testing results is notably diminished compared to traditional partial differential equation (PDE) solvers (reduced by about 3 to 12 times). To demonstrate model robustness, synthetic noise is intentionally introduced into the training data. The results not only enhance our understanding of the complex dynamics of the OVEP but also highlight the effectiveness of the proposed PINN framework in capturing and forecasting the dynamic behaviors of advanced plate structures. This research presents a promising potential for addressing dynamic problems in the fields of aerospace, civil, and mechanical engineering.