Article
Engineering, Mechanical
Jinwoo Im, Felipe P. J. de Barros, Sami F. Masri
Summary: We propose a stochastic optimization framework to identify governing equations in multi-physics systems. This framework combines prior knowledge of the target system's physics and observed data to discover partial differential equations (PDEs) through evolutionary processes. The proposed method captures the spatiotemporal dynamics of the physical system and achieves significant computational speedup through novel modifications.
NONLINEAR DYNAMICS
(2023)
Article
Engineering, Multidisciplinary
Joshua Gasick, Xiaoping Qian
Summary: This paper presents a physics-based machine learning approach for solving parameterized PDEs. By combining isogeometric representation and deep learning, the approach uses non-uniform rational B-spline basis functions to represent the physical domain and the PDE solution, and employs a deep feed-forward neural network to predict the NURBS solution coefficients. The method can learn PDE solutions for various parametric variables without requiring labeled data.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Engineering, Multidisciplinary
Kazem Meidani, Amir Barati Farimani
Summary: This study introduces a machine learning method to discover terms in partial differential equations from spatiotemporal data. By extracting robust physical features from data samples, the method can effectively represent the behaviors imposed by each mathematical term. In comparison to previous models, this approach allows for the discovery of 2D equations with different orders of time derivatives and identification of new underlying physics.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Siping Tang, Xinlong Feng, Wei Wu, Hui Xu
Summary: In this paper, the authors propose a simplified neural network called polynomial interpolation physics-informed neural networks (PI-PINN) to solve nonlinear partial differential equations. By utilizing orthogonal polynomials to construct the neural network, the PI-PINN structure is shown to be effective in solving these equations. Numerical experiments and investigations on reverse problems demonstrate the accuracy and efficiency of the proposed approach.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Shuyan Shi, Ding Liu, Ruirui Ji, Yuchao Han
Summary: This paper proposes a new physics-informed neural network method, which solves the problem of solving partial differential equations by introducing spatial-temporal coefficients and balance parameters. A two-stage training process is used to improve convergence, effectively and accurately solving problems such as unbalanced loss functions, hidden errors, and conduction errors.
NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Ali C. Bekar, Erdogan Madenci
Summary: This study introduces an approach to identify significant terms in PDEs based on measured data using linear regression model, PDDO, and sparse linear regression learning algorithm. The solution is achieved through Douglas-Rachford algorithm with regularization, demonstrating effectiveness in handling challenging nonlinear PDEs.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Engineering, Multidisciplinary
Vahidullah Tac, Manuel K. Rausch, Francisco Sahli Costabal, Adrian Buganza Tepole
Summary: We develop a fully data-driven model of anisotropic finite viscoelasticity using neural ordinary differential equations. The model replaces the Helmholtz free energy function and the dissipation potential with data-driven functions that satisfy physics-based constraints. This approach allows modeling of viscoelastic behavior under arbitrary loads in three-dimensions and large deviations from thermodynamic equilibrium. The data-driven nature of the model provides flexibility in modeling a wide class of materials and outperforms traditional models of viscoelasticity.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Computer Science, Interdisciplinary Applications
Apostolos F. Psaros, Xuhui Meng, Zongren Zou, Ling Guo, George Em Karniadakis
Summary: Neural networks have revolutionized computational methods by effectively solving challenging problems that cannot be solved using traditional methods. However, quantifying errors and uncertainties in neural network-based inference is more complex than in traditional methods. This study presents a comprehensive framework for effectively and efficiently quantifying total uncertainty in neural networks, including uncertainty modeling, solution methods, and evaluation metrics. Furthermore, an open-source Python library called NeuralUQ is developed to facilitate the deployment of uncertainty quantification in scientific machine learning research and practice.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Quantum Science & Technology
Caio B. D. Goes, Thiago O. O. Maciel, Giovani G. G. Pollachini, Juan P. L. C. Salazar, Rafael G. G. Cuenca, Eduardo I. I. Duzzioni
Summary: This paper proposes a hybrid algorithm based on machine learning and quantum ensemble learning to find an approximate solution to a partial differential equation with good precision and favorable scaling in the required number of qubits. The classical component trains multiple regressors capable of approximately solving the equation using machine learning, while the quantum component adapts the QBoost algorithm to build an ensemble of classical learners. The algorithm is successfully applied to solve the 1D Burgers' equation with viscosity, demonstrating the improved solutions compared to classical weak-learners, and implemented on D-Wave Systems with good performance compared to other methods.
QUANTUM INFORMATION PROCESSING
(2023)
Article
Computer Science, Artificial Intelligence
Somdatta Goswami, Katiana Kontolati, Michael D. D. Shields, George Em Karniadakis
Summary: This paper presents a transfer learning framework for task-specific learning based on Deep Operator Network (DeepONet), addressing conditional shifts in transfer learning scenarios such as nonlinear partial differential equations.
NATURE MACHINE INTELLIGENCE
(2022)
Article
Mathematics, Applied
Salvatore Cuomo, Vincenzo Schiano Di Cola, Fabio Giampaolo, Gianluigi Rozza, Maziar Raissi, Francesco Piccialli
Summary: Physics-Informed Neural Networks (PINN) are a type of neural network that incorporates model equations, such as partial differential equations, as a component. PINNs have been used to solve various types of equations, including fractional equations and stochastic partial differential equations. Current research focuses on optimizing PINN through different aspects, such as activation functions, gradient optimization techniques, neural network structures, and loss function structures. Despite the demonstrated feasibility of PINN in certain cases compared to traditional numerical techniques, there are still unresolved theoretical issues.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Computer Science, Artificial Intelligence
Robert Stephany, Christopher Earls
Summary: A new approach using two neural networks and sparse regression algorithm is introduced for discovering predictive models of complex physical systems, showing robustness in dealing with noisy and limited measurements.
Article
Computer Science, Interdisciplinary Applications
I Boureima, V Gyrya, J. A. Saenz, S. Kurien, M. Francois
Summary: This article presents a methodology for calibrating parametric ordinary and partial differential equation models using Neural Networks and back-propagation. The methodology is tested on a Reynolds-averaged Navier-Stokes turbulence closure model using ground truth data from direct numerical simulations. The dynamic approach, which minimizes a loss function through gradient descent, proves to be more accurate and stable than the static approach. After optimization, the dynamic approach shows a significant improvement over manually calibrated parameters.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Engineering, Multidisciplinary
Weiheng Zhong, Hadi Meidani
Summary: This paper proposes a new class of physics-informed neural networks, called the Physics-Informed Variational Auto-Encoder (PI-VAE), for solving stochastic differential equations (SDEs) or inverse problems involving SDEs. The PI-VAE integrates a variational autoencoder (VAE) with the governing equations, using automatic differentiation to calculate derivatives and the Maximum Mean Discrepancy (MMD) as the loss function. Numerical experiments demonstrate the satisfactory accuracy and efficiency of the proposed method in approximating stochastic processes and solving forward, inverse, and mixed problems related to SDEs.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Engineering, Multidisciplinary
Yasen Wang, Huazhen Fang, Junyang Jin, Guijun Ma, Xin He, Xing Dai, Zuogong Yue, Cheng Cheng, Hai-Tao Zhang, Donglin Pu, Dongrui Wu, Ye Yuan, Jorge Goncalves, Juergen Kurths, Han Ding
Summary: This study presents a novel framework for identifying stochastic differential equations (SDEs) by leveraging sparse Bayesian learning (SBL) technique. The framework automatically searches for a parsimonious representation from the space of candidate basis functions using SBL and formulates the linear regression problem for the discovery of SDEs in an efficient way. The effectiveness of the proposed framework is demonstrated using real and simulated data.
Article
Green & Sustainable Science & Technology
S. W. Funke, S. C. Kramer, M. D. Piggott
Article
Engineering, Biomedical
Simon Wolfgang Funke, Magne Nordaas, Oyvind Evju, Martin Sandve Alnaes, Kent Andre Mardal
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING
(2019)
Article
Oncology
Xiaoran Lai, Oliver M. Geier, Thomas Fleischer, Ystein Garred, Elin Borgen, Simon W. Funke, Surendra Kumar, Marie E. Rognes, Therese Seierstad, Anne-Lise Borresen-Dale, Vessela N. Kristensen, Olav Engebraaten, Alvaro Kohn-Luque, Arnoldo Frigessi
Article
Engineering, Multidisciplinary
Jrgen S. Dokken, August Johansson, Andre Massing, Simon W. Funke
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2020)
Article
Engineering, Biomedical
Xiaoran Lai, Hakon A. Tasken, Torgeir Mo, Simon W. Funke, Arnoldo Frigessi, Marie E. Rognes, Alvaro Kohn-Luque
Summary: Mathematical modeling and simulation provide a promising approach to personalized cancer medicine, but face challenges due to the complexity, heterogeneity, and multi-scale nature of cancer. Hybrid cellular automata, coupling discrete cell-based models with continuous models, offer a powerful method to mimic biological complexity across different scales. The development of parallel algorithms to link stochastic CA with differential equations shows potential for efficient computation in personalized therapy simulations.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING
(2022)
Article
Computer Science, Interdisciplinary Applications
Tian Liang, Lin Fu
Summary: In this work, a new shock-capturing framework is proposed based on a new candidate stencil arrangement and the combination of infinitely differentiable non-polynomial RBF-based reconstruction in smooth regions with jump-like non-polynomial interpolation for genuine discontinuities. The resulting scheme achieves high order accuracy and resolves genuine discontinuities with sub-cell resolution.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Lukas Lundgren, Murtazo Nazarov
Summary: In this paper, a high-order accurate finite element method for incompressible variable density flow is introduced. The method addresses the issues of saddle point system and stability problem through Schur complement preconditioning and artificial compressibility approaches, and it is validated to have high-order accuracy for smooth problems and accurately resolve discontinuities.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Gabriele Ciaramella, Laurence Halpern, Luca Mechelli
Summary: This paper presents a novel convergence analysis of the optimized Schwarz waveform relaxation method for solving optimal control problems governed by periodic parabolic PDEs. The analysis is based on a Fourier-type technique applied to a semidiscrete-in-time form of the optimality condition, which enables a precise characterization of the convergence factor at the semidiscrete level. The behavior of the optimal transmission condition parameter is also analyzed in detail as the time discretization approaches zero.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jonas A. Actor, Xiaozhe Hu, Andy Huang, Scott A. Roberts, Nathaniel Trask
Summary: This article introduces a scientific machine learning framework that uses a partition of unity architecture to model physics through control volume analysis. The framework can extract reduced models from full field data while preserving the physics. It is applicable to manifolds in arbitrary dimension and has been demonstrated effective in specific problems.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Nozomi Magome, Naoki Morita, Shigeki Kaneko, Naoto Mitsume
Summary: This paper proposes a novel strategy called B-spline based SFEM to fundamentally solve the problems of the conventional SFEM. It uses different basis functions and cubic B-spline basis functions with C-2-continuity to improve the accuracy of numerical integration and avoid matrix singularity. Numerical results show that the proposed method is superior to conventional methods in terms of accuracy and convergence.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Timothy R. Law, Philip T. Barton
Summary: This paper presents a practical cell-centred volume-of-fluid method for simulating compressible solid-fluid problems within a pure Eulerian setting. The method incorporates a mixed-cell update to maintain sharp interfaces, and can be easily extended to include other coupled physics. Various challenging test problems are used to validate the method, and its robustness and application in a multi-physics context are demonstrated.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Xing Ji, Fengxiang Zhao, Wei Shyy, Kun Xu
Summary: This paper presents the development of a third-order compact gas-kinetic scheme for compressible Euler and Navier-Stokes solutions, constructed particularly for an unstructured tetrahedral mesh. The scheme demonstrates robustness in high-speed flow computation and exhibits excellent adaptability to meshes with complex geometrical configurations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Alsadig Ali, Abdullah Al-Mamun, Felipe Pereira, Arunasalam Rahunanthan
Summary: This paper presents a novel Bayesian statistical framework for the characterization of natural subsurface formations, and introduces the concept of multiscale sampling to localize the search in the stochastic space. The results show that the proposed framework performs well in solving inverse problems related to porous media flows.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jacob Rains, Yi Wang, Alec House, Andrew L. Kaminsky, Nathan A. Tison, Vamshi M. Korivi
Summary: This paper presents a novel method called constrained optimized DMD with Control (cOptDMDc), which extends the optimized DMD method to systems with exogenous inputs and can enforce the stability of the resulting reduced order model (ROM). The proposed method optimally places eigenvalues within the stable region, thus mitigating spurious eigenvalue issues. Comparative studies show that cOptDMDc achieves high accuracy and robustness.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Andrea La Spina, Jacob Fish
Summary: This work introduces a hybridizable discontinuous Galerkin formulation for simulating ideal plasmas. The proposed method couples the fluid and electromagnetic subproblems monolithically based on source and employs a fully implicit time integration scheme. The approach also utilizes a projection-based divergence correction method to enforce the Gauss laws in challenging scenarios. Numerical examples demonstrate the high-order accuracy, efficiency, and robustness of the proposed formulation.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Junhong Yue, Peijun Li
Summary: This paper proposes two numerical methods (IP-FEM and BP-FEM) to study the flexural wave scattering problem of an arbitrary-shaped cavity on an infinite thin plate. These methods successfully decompose the fourth-order plate wave equation into the Helmholtz and modified Helmholtz equations with coupled conditions on the cavity boundary, providing an effective solution to this challenging problem.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
William Anderson, Mohammad Farazmand
Summary: We develop fast and scalable methods, called RONS, for computing reduced-order nonlinear solutions. These methods have been proven to be highly effective in tackling challenging problems, but become computationally prohibitive as the number of parameters grows. To address this issue, three separate methods are proposed and their efficacy is demonstrated through examples. The application of RONS to neural networks is also discussed.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Marco Caliari, Fabio Cassini
Summary: In this paper, a second order exponential scheme for stiff evolutionary advection-diffusion-reaction equations is proposed. The scheme is based on a directional splitting approach and uses computation of small sized exponential-like functions and tensor-matrix products for efficient implementation. Numerical examples demonstrate the advantage of the proposed approach over state-of-the-art techniques.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Sebastiano Boscarino, Seung Yeon Cho, Giovanni Russo
Summary: This work proposes a high order conservative semi-Lagrangian method for the inhomogeneous Boltzmann equation of rarefied gas dynamics. The method combines a semi-Lagrangian scheme for the convection term, a fast spectral method for computation of the collision operator, and a high order conservative reconstruction and a weighted optimization technique to preserve conservative quantities. Numerical tests demonstrate the accuracy and efficiency of the proposed method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jialei Li, Xiaodong Liu, Qingxiang Shi
Summary: This study shows that the number, centers, scattering strengths, inner and outer diameters of spherical shell-structured sources can be uniquely determined from the far field patterns. A numerical scheme is proposed for reconstructing the spherical shell-structured sources, which includes a migration series method for locating the centers and an iterative method for computing the inner and outer diameters without computing derivatives.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)