Physics-informed neural networks (PINNs) have recently emerged as a promising paradigm for simulating three-dimensional fluid-structure interaction (FSI) in cardiovascular systems. Building on our Integral Conservation Physics-Informed Neural Networks (ICPINNs) framework for patient-specific hemodynamics [Liu Y, et al., Comput. Phys. Commun., 2025, 109569], this study develops an extended framework for elastic vessels, termed E-ICPINNs. E-ICPINNs adopts a dual-subnetwork design, in which separate subnetworks learn the fluid variables and structural variables, respectively. The incompressible Navier-Stokes equations in an arbitrary Lagrangian-Eulerian formulation and vessel wall mechanics are embedded as meshless physics-informed constraints, thereby enforcing FSI coupling in deformable vascular domains. The coupled problem is further formulated as a multi-objective optimization task, and a sequential-alternative training algorithm is developed by combining regularization-based pretraining with alternating optimization over subsets of physics-informed loss terms, which improves training stability and convergence under strong coupling. In addition, an adaptive SiLU activation function is incorporated to preserve smooth differentiability across coupled fields and to mitigate vanishing-gradient effects in deep networks. Comprehensive numerical experiments, together with systematic comparisons of network depth/width, activation functions, and representative architectures, demonstrate that E-ICPINNs achieves accurate and robust predictions with favorable accuracy-efficiency trade-offs. The proposed framework provides an effective meshless computational approach for clinically relevant vascular FSI simulations.

The numerical simulation of blood flow in the patient-specific thoracic aorta not only accurately reproduces personalized hemodynamic characteristics but also provides robust data support for the diagnosis and treatment of vascular diseases. This study advances the numerical simulation of blood flow in patient-specific thoracic aortas by extending our previously developed Integral Conservation Physics-Informed Neural Networks (ICPINNs) framework (Liu et al., 2025) from steady-state to transient flow problems. The ICPINNs method leverages the integral conservation form of the nonlinear Navier-Stokes equations, incorporating residual terms derived from both governing equations and training data, with Monte Carlo integration employed for integrals. We address two main classes of aortas: (1) unsupervised learning for anomalous branching of the aorta, and (2) integration of sparse velocity measurements for geometrically complex healthy and pathological full thoracic aortas. Furthermore, we conduct the first systematic comparison of different neural network architectures for real-world transient aortic flows, assessing their computational efficiency and accuracy against conventional numerical solutions. Numerical results demonstrate that fully-connected neural networks within the ICPINNs framework achieves optimal performance for healthy aortas, while more sophisticated architectures such as the Deep Galerkin Method prove superior for modeling complex pathologies like Marfan syndrome-associated aneurysms, despite increased computational costs. This work represents an important step toward personalized hemodynamic modeling, offering clinically relevant insights that could enhance diagnostic precision and therapeutic planning for cardiovascular diseases.

Blood flow modeling can improve our understanding of vascular pathologies, assist in designing more effective drug delivery systems, and aid in developing safe and effective medical devices. Physics-informed neural networks (PINN) have been used to simulate blood flow by encoding the nonlinear Navier-Stokes equations and training data into the neural network. However, noninvasive, real-time and accurate acquisition of hemodynamics data remains a challenge for current invasive detection and simulation algorithms. In this paper, we propose an integral conservation physics-informed neural networks (ICPINN) with adaptive activation functions to accurately predict the velocity, pressure, and wall shear stress (WSS) based on patient-specific vessel geometries without relying on any simulation data. To achieve unsupervised learning, loss function incorporates mass flow rate residuals derived from the mass conservation law, significantly enhancing the precision and effectiveness of the predictions. Moreover, a detailed comparative analysis of various weighting coefficient selection strategies and activation functions is performed, which ultimately identifies the optimal configuration for 3D blood flow simulations that achieves the lowest relative error. Numerical results demonstrate that the proposed ICPINN framework enables accurate prediction of blood flow in realistic cardiovascular geometry, and that mass flow rate is essential for complex structures, such as bifurcations, U-bend, stenosis, and aneurysms, offering potential applications in medical diagnostics and treatment planning.

We present a high-order accurate Godunov method for solving both modified and pure Chaplygin gas dynamics equations. The approach involves analyzing and applying an exact Riemann solver to construct the Godunov fluxes for both types of gas dynamics. To effectively resolve discontinuities in numerical simulations, we employ a classical weighted essentially nonoscillatory (WENO) scheme and a monotonicity-preserving scheme for the reconstruction of cell averages. A series of numerical tests in both one and two dimensions are conducted to demonstrate the accuracy and robustness of the proposed numerical scheme. Additionally, we examine the behavior of numerical solutions for the modified Chaplygin gas equations. Our results confirm that as the parameter A in the modified Chaplygin gas equation of state approaches zero, the solutions of the modified Chaplygin gas equations converge to those of the pure Chaplygin gas equations, potentially exhibiting delta shock waves.

This paper presents a high-order, finite volume, monotonicity-preserving WENO method that adheres to physical constraints (specifically, density ρ > 0 and internal energy e > 1/2ρ ^2 ) for solving the Chaplygin gas dynamic equations. Due to the presence of negative pressure, the solutions may include delta waves, which pose significant numerical approximation challenges. Furthermore, negative pressure exacerbates the nonlinearity in the constraint on internal energy, leading to the invalidity of the typically expected Lax–Friedrichs (LF) splitting property. This complexity makes the design and analysis of our physical-constraint-preserving (PCP) schemes nontrivial. To address these challenges, we employ a novel geometric quasilinearization (GQL) approach recently proposed in [K. Wu C.-W. Shu, SIAM Rev., 65(4):1031–1073, 2023]. Using GQL, we derive an equivalent linear representation of the admissible state set through the introduction of appropriate auxiliary variables. As alternatives to the invalid LF splitting property, we discover several critical inequalities and generalized LF splitting properties, thereby laying the groundwork for analyzing and designing PCP schemes. Thanks to these properties, PCP analysis is significantly simplified into seeking a summation of positive terms. To achieve high resolution for contact discontinuities and delta waves, we present a robust, high-order, monotonicity-preserving WENO method for the Chaplygin gas dynamics. The PCP property of this method is rigorously proven under conditions enforced by the PCP limiter. We conduct numerical experiments in one and two dimensions to evaluate the accuracy and robustness of our method. Our PCP method effectively captures the behavior of delta waves and contact discontinuities, even in scenarios with low density and internal energy.

Physics-informed neural networks (PINN) architectures have been recently explored to accelerate hemodynamics simulations by leveraging mathematical models for blood flow and empirical data. In this paper, a variable separated physics-informed neural networks based on adaptive weighted loss functions (AW-vsPINN) is developed for blood flow model in arteries. In particular, sub-neural networks are proposed to separately predict the unknown scalar state variables by sharing the same input layer. The AW-vsPINN adaptively adjusts the weights of loss terms by the minmax algorithm, which will be updated synchronously along with the network parameters and can balance the contributions of different loss terms during training. The two-stage optimization is implemented to train the neural networks. Specifically, the Adam optimizer is iterated for initial steps with the learning rate generated by the inverse time decay scheduler, and then the L-BFGS optimizer continues to train until the loss converges. Numerical results illustrate that the AW-vsPINN can remarkably improve prediction accuracy and enhance the ability of generalization compared to the conventional PINN. The proposed AW-vsPINN framework has high potential in predicting the blood flow information in cardiovascular disease.

Physics-informed neural networks (PINN) have lately become a research hotspot in the interdisciplinary field of machine learning and computational mathematics thanks to the flexibility in tackling forward and inverse problems. In this work, we explore the generality of the PINN training algorithm for solving Hamilton-Jacobi equations, and propose physics-informed neural networks based on adaptive weighted loss functions (AW-PINN) that is trained to solve unsupervised learning tasks with fewer training data while physical information constraints are imposed during the training process. To balance the contributions from different constrains automatically, the AW-PINN training algorithm adaptively update the weight coefficients of different loss terms by using the logarithmic mean to avoid additional hyperparameter. Moreover, the proposed AW-PINN algorithm imposes the periodicity requirement on the boundary condition and its gradient. The fully connected feedforward neural networks are considered and the optimizing procedure is taken as the Adam optimizer for some steps followed by the L-BFGS-B optimizer. The series of numerical experiments illustrate that the proposed algorithm effectively achieves noticeable improvements in predictive accuracy and the convergence rate of the total training error, and can approximate the solution even when the Hamiltonian is nonconvex. A comparison between the proposed algorithm and the original PINN algorithm for Hamilton-Jacobi equations indicates that the proposed AW-PINN algorithm can train the solutions more accurately with fewer iterations.

This paper presents a highly robust third-order accurate finite volume weighted essentially non-oscillatory (WENO) method for special relativistic hydrodynamics on unstructured triangular meshes. We rigorously prove that the proposed method is physical-constraintpreserving (PCP), namely, always preserves the positivity of the pressure and the rest-mass density as well as the subluminal constraint on the fluid velocity. The method is built on a highly efficient compact WENO reconstruction on unstructured meshes, a simple PCP limiter, the provably PCP property of the Harten-Lax-van Leer flux, and third-order strong-stability-preserving time discretization. Due to the relativistic effects, the primitive variables (namely, the rest-mass density, velocity, and pressure) are highly nonlinear implicit functions in terms of the conservative variables, making the design and analysis of our method nontrivial. To address the difficulties arising from the strong nonlinearity, we adopt a novel quasilinear technique for the theoretical proof of the PCP property. Three provable convergence-guaranteed iterative algorithms are also introduced for the robust recovery of primitive quantities from admissible conservative variables. We also propose a slight modification to an existing WENO reconstruction to ensure the scaling invariance of the nonlinear weights and thus to accommodate the homogeneity of the evolution operator, leading to the advantages of the modified WENO reconstruction in resolving multi-scale wave structures. Extensive numerical examples are presented to demonstrate the robustness, expected accuracy, and high resolution of the proposed method.

This paper extends the second-order accurate BGK finite volume schemes for the ultra-relativistic flow simulations [5] to the 1D and 2D special relativistic hydrodynamics with the Synge equation of state. It is shown that such 2D schemes are very time-consuming due to the moment integrals (triple integrals) so that they are no longer practical. In view of this, the simplified BGK (sBGK) schemes are presented by removing some terms in the approximate nonequilibrium distribution at the cell interface for the BGK scheme without loss of accuracy. They are practical because the moment integrals of the approximate distribution can be reduced to the single integrals by some coordinate transformations. The relations between the left and right states of the shock wave, rarefaction wave, and contact discontinuity are also discussed, so that the exact solution of the 1D Riemann problem could be derived and used for the numerical comparisons. Several numerical experiments are conducted to demonstrate that the proposed gas-kinetic schemes are accurate and stable. A comparison of the sBGK schemes with the BGK scheme in one dimension shows that the former performs almost the same as the latter in terms of the accuracy and resolution, but is much more efficiency.

Distributed-order fractional differential equations, where the differential order is distributed over a range of values rather than being just a fixed value as it is in the classical differential equations, offer a powerful tool to describe multi-physics phenomena. In this article, we develop and analyze an efficient finite difference/generalized Hermite spectral method for the distributed-order time-fractional reaction-diffusion equation on one-, two-, and three-dimensional unbounded domains. Considering the Gauss-Legendre quadrature rule for the distributed integral term in temporal direction, we first approximate the original distributed-order time-fractional problem by the multi-term time-fractional differential equation. Then, we apply the L2-1σ formula for the discretization of the multi-term Caputo fractional derivatives. Moreover, we employ the generalized Hermite functions with scaling factor for the spectral approximation in space. The detailed implementations of the method are presented for one-, two-, and three-dimensional cases of the fractional problem. The stability and convergence of the method are strictly established, which shows that the proposed method is unconditionally stable and convergent with second-order accuracy in time. In addition, the optimal error estimate is derived for the space approximation. Finally, we perform numerical examples to support the theoretical claims.

This paper presents second-order accurate genuine BGK schemes in the framework of finite volume method for the ultra-relativistic flows. Different from the existing kinetic flux-vector splitting (KFVS) or BGK-type schemes for the ultra-relativistic Euler equations, the present schemes are derived from the analytical solution of the Anderson–Witting model, which is given for the first time and includes the “genuine” particle collisions in the gas transport process. The proposed schemes for the ultra-relativistic viscous flows are also developed and two examples of ultra-relativistic viscous flow are designed. Several 1D and 2D numerical experiments are conducted to demonstrate that the proposed schemes not only are accurate and stable in simulating ultra-relativistic inviscid and viscous flows, but also have higher resolution at the contact discontinuity than the KFVS or BGK-type schemes.

A model singularly perturbed convection–diffusion problem in two space dimensions is considered. The problem is solved by a streamline diffusion finite element method (SDFEM) that uses piecewise bilinear finite elements on a Shishkin mesh. We prove that the method is convergent, independently of the diffusion parameter ε, with a pointwise accuracy of almost order 7/4 away from the characteristic layers. Numerical experiments support these theoretical results.