In this paper, we propose a parameter-robust preconditioner for the coupled Stokes-Darcy problem equipped with various boundary conditions, enforcing the mass conservation at the interface via a Lagrange multiplier. We rigorously establish that the coupled system is well-posed with respect to physical parameters and mesh size and provides a framework for constructing parameterrobust preconditioners. Furthermore, we analyze the convergence behavior of the minimal residual method in the presence of small outlier eigenvalues linked to specific boundary conditions, which can lead to slow convergence or stagnation. To address this issue, we employ deflation techniques to accelerate the convergence. Finally, numerical experiments confirm the effectiveness and robustness of the proposed approach.

Gradient methods are widely used in optimization problems. In practice, while the smoothness parameter can be estimated utilizing techniques such as backtracking, estimating the strong convexity parameter remains a challenge; moreover, even with the optimal parameter choice, convergence can be slow. In this work, we propose a framework for dynamically adapting the step size and momentum parameters in first-order gradient methods for the optimization problem, without prior knowledge of the strong convexity parameter. The main idea is to use the geometric average of the ratios of successive residual norms as an empirical estimate of the upper bound on the convergence rate, which in turn allows us to adaptively update the algorithm parameters. The resulting algorithms are simple to implement, yet efficient in practice, requiring only a few additional computations on existing information. The proposed adaptive gradient methods are shown to converge at least as fast as gradient descent for quadratic optimization problems. Numerical experiments on both quadratic and nonlinear problems validate the effectiveness of the proposed adaptive algorithms. The results show that the adaptive algorithms are comparable to their counterparts using optimal parameters, and in some cases, they capture local information and exhibit improved performance.

Protein function prediction from multiplex protein-protein association networks is a crucial approach to extending functional annotation. Current methods use embeddings of the heterogeneous network data that aim to place related proteins near each other in embedding space. However, such embeddings suffer from spurious protein proximity as well, reducing function prediction accuracy. Because heterogeneous input networks often have very different structures, it is hard to confidently declare proteins to be dissimilar using the network structure or the resulting embeddings. Here we address this problem with REPEL, a function prediction tool using a random graph augmentation method that applies a uniform weak force to push nodes apart. We assess this method on simulated networks with planted overlapping communities, as well as on real multiplex yeast and E.coli protein association networks. Surprisingly, we find that this method consistently improves protein function prediction over competing methods Mashup, deepNF, and BIONIC. The random repelling nature of the augmented graphs has a denoising effect on the learning process, distancing node pairs with spurious proximity while preserving true functional connections, thus increasing robustness. This graph augmentation principle may generalize to denoising and improving robustness in other graph-based learning algorithms.

For autoregressive modeling of chaotic dynamical systems over long time horizons, the stability of both training and inference is a major challenge in building scientific foundation models. We present a hybrid technique in which an autoregressive transformer is embedded within a novel shooting-based mixed finite element scheme, exposing topological structure that enables provable stability. For forward problems, we prove preservation of discrete energies, while for training we prove uniform bounds on gradients, provably avoiding the exploding gradient problem. Combined with a vision transformer, this yields latent tokens admitting structure-preserving dynamics. We outperform modern foundation models with a 65× reduction in model parameters and long-horizon forecasting of chaotic systems. A "mini-foundation" model of a fusion component shows that 12 simulations suffice to train a real-time surrogate, achieving a 9,000× speedup over particle-in-cell simulation.

In recent years, quantum linear system algorithms have been applied to partial differential equations (PDEs), particularly in high-dimensional settings, demonstrating an exponential speedup in dimension. Concurrently, randomized and quantum-inspired classical linear solvers have emerged, showing computational complexity comparable to their quantum counterparts in many application areas. In this paper, we investigate the applicability of these quantum-inspired classical algorithms to PDEs. We provide both upper and lower bounds on their computational complexity, proving that these methods cannot achieve exponential speedup in dimension for discretizations of high-dimensional Poisson problems. Our theoretical findings definitively demonstrate that quantum-inspired classical algorithms are not competitive with quantum algorithms for solving PDEs, confirming that quantum methods retain a significant advantage for high-dimensional problems.

In this work, we develop a nonconforming finite element method for the three-field Biot model, where the variables are displacement, Darcy velocity, and pore pressure. The discretization employs the lowest-order Crouzeix-Raviart (CR) element for both displacement and Darcy velocity, and piecewise constant element for pressure. We establish the well-posedness of the discrete problem with respect to a carefully chosen weighted norm, ensuring robustness with respect to both discretization and physical parameters. Furthermore, we prove optimal convergence of the proposed scheme. To improve computational efficiency, we introduce a reduced-order CR method based on the proper orthogonal decomposition (POD) technique. Numerical experiments are provided to verify the theoretical convergence rates and to demonstrate the effectiveness of the reduced-order approach.

We propose a unified four-dimensional (4D) spatiotemporal formulation for time-dependent convection-diffusion problems that preserves underlying physical structures. By treating time as an additional space-like coordinate, the evolution problem is reformulated as a stationary convection-diffusion equation on a 4D space-time domain. Using exterior calculus, we extend this framework to the full family of convection-diffusion problems posed on H(grad), H(curl), and H(div). The resulting formulation is based on a 4D Hodge-Laplacian operator with a spatiotemporal diffusion tensor and convection field, augmented by a small temporal perturbation to ensure nondegeneracy. This formulation naturally incorporates fundamental physical constraints, including divergence-free and curl-free conditions. We further introduce an exponentially-fitted 4D spatiotemporal flux operator that symmetrizes the convection-diffusion operator and enables a well-posed variational formulation. Finally, we prove that the temporally-perturbed formulation converges to the original time-dependent convection-diffusion model as the perturbation parameter tends to zero.

This paper develops a novel residual-based a posteriori error estimator for a pressure-robust Galerkin finite-element method for the Stokes equations. Recent studies show that pressure-robust Stokes solvers outperform non-robust solvers, especially for problems with small viscosity or singular pressure unknowns. We base our analysis on a pressure-robust scheme introduced in [22], which approximates velocity with piecewise continuous P_1 functions enriched by RT _0 functions and pressure with discontinuous P_0 functions. While previous a priori error estimates show velocity error independence from pressure and viscosity, we propose an efficient and reliable a posteriori error estimator that maintains this robustness. Theoretical analysis demonstrates the equivalence between the exact H^1 velocity error and the fully computable proposed estimator. Unlike traditional a posteriori estimates that include pressure contributions, our approach depends solely on velocity-related terms. This enables the estimator to distinguish between velocity and pressure singularities and guide mesh refinement for velocity improvement. Numerical results validate the efficiency and reliability of the estimate, and the developed error indicator is applied in an automatic, self-adaptive mesh refinement process to capture velocity singularities.

An alternative to the fully implicit or monolithic methods used for the solution of the coupling of fluid flow and deformation in porous media is a sequential approach in which the fully coupled system is broken into subproblems (flow and mechanics problems) that are solved one after the other. This fully explicit coupling approach is a very simple scheme which allows much flexibility in the implementation and has a lower computational cost, making it quite attractive in practice since smaller linear systems need to be solved in order to obtain the solution for the whole coupled poroelastic system. Due to the appealing advantages of these methods, intensive research is currently being carried out in this direction, as in the present work. Although the application of this type of method is very common in practice, there exist only a few works devoted to their theoretical analysis. In this work, we consider the so-called explicit fixed-stress split scheme, which consists of solving the flow problem first with time-lagging the displacement term, followed by the solution of the mechanics problem. To the best of our knowledge, we provide the first convergence analysis of the explicit fixed-stress split scheme for Biot's equations. In particular, we prove that this algorithm is optimally convergent if the considered finite element discretization satisfies an inf-sup condition. In addition, with the aim of designing the simplest scheme for solving Biot's model, we also propose a similar decoupled algorithm for piecewise linear finite elements for both variables which arises from the novel stabilization recently proposed in , and is demonstrated to be optimally convergent.

The Arnoldi process provides an efficient framework for approximating functions of a matrix applied to a vector, i.e., of the form f(M)𝐛, by repeated matrix-vector multiplications. In this paper, we derive an a priori error estimate for approximating the action of a matrix square root using the Arnoldi process, where the integral representation of the error is reformulated in terms of the error for solving the linear system M𝐱=𝐛. The results extend the error analysis of the Lanczos method for Hermitian matrices in [Chen et al., SIAM J. Matrix Anal. Appl., 2022] to non-Hermitian cases. Furthermore, to make the method applicable to large-scale problems, we assume that the matrices are preprocessed utilizing data-sparse approximations preserving positive definiteness, and then establish a refined error bound in this setting. The numerical results on matrices with different structures demonstrate that our theoretical analysis yields a reliable upper bound. Finally, simulations on large-scale matrices arising in particulate suspensions validate the effectiveness and practicality of the approach.

In this paper, we propose a new deep learning method for the nonlinear Poisson-Boltzmann problems with applications in computational biology. To tackle the discontinuity of the solution, e.g., across protein surfaces, we approximate the solution by a piecewise mesh-free neural network that can capture the dramatic change in the solution across the interface. The partial differential equation problem is first reformulated as a least-squares physics-informed neural network (PINN)-type problem and then discretized to an objective function using mean squared error via sampling. The solution is obtained by minimizing the designed objective function via standard training algorithms such as the stochastic gradient descent method. Finally, the effectiveness and efficiency of the neural network are validated using complex protein interfaces on various manufactured functions with different frequencies.

We consider the Stokes-Darcy coupled problem, which models the interaction between free-flow and porous medium flow. By enforcing the normal flux continuity interface condition directly within the finite-element spaces, we establish unified well-posedness results for the coupled system under various boundary condition scenarios. Using the operator preconditioning framework, we develop a parameter-robust preconditioner that avoids the use of fractional operators. Numerical experiments employing both H(div)-conforming and nonconforming finite-element methods are presented to confirm the theoretical findings and demonstrate the robustness of the proposed block preconditioners with respect to the physical parameters and mesh size.

In this work, we present a new stabilization method aimed at removing spurious oscillations in the pressure approximation of Biot's model for poroelasticity with low permeabilities and/or small time steps. We consider different finite-element discretizations and illustrate how not only does such a stabilized scheme provide numerical solutions that are free of nonphysical oscillations, but it also allows one to iterate the fluid and mechanics problems in a fashion similar to the well-known fixed-stress split method. The resulting iterative solution method is convergent without adding extra stabilization terms. Finally, we present numerical results illustrating the robust behavior of both the stabilization and iterative solver with respect to the physical and discretization parameters of the model.

We develop multilevel methods for interface-driven multiphysics problems that can be coupled across dimensions and where complexity and strength of the interface coupling deteriorates the performance of standard methods. We focus on solvers based on aggregation-based algebraic multigrid methods with custom smoothers that preserve the coupling information on each coarse level. We prove that with the proper choice of subspace splitting we obtain uniform convergence in discretization and physical parameters in the two-level setting. Additionally, we show parameter robustness and scalability with regards to number of the degrees of freedom of the system on several numerical examples related to the biophysical processes in the brain, namely the electric signalling in excitable tissue modeled by bidomain, EMI and reduced EMI equations.

The pioneering and still popular IsoRank method of Singh, Xu, and Berger for global alignment of two protein-protein interaction networks across species was introduced at Recomb in 2007, and was awarded the Recomb test of time award in 2019. However, with the availability of increasing amounts of experimental data the number of edges in the networks to align has grown considerably, making running IsoRank unfeasible on these networks without access to substantial computational resources. In this paper, we develop a new IsoRank approximation that exploits the mathematical properties of IsoRank’s linear system to solve the problem in quadratic time with respect to the maximum size of the two PPI networks. We further propose a refinement to this initial approximation so that the updated result is even closer to the original IsoRank formulation while remaining computationally inexpensive. In experiments on synthetic and real PPI networks with various proposed metrics to measure alignment quality, we find the results of our approximate IsoRank are nearly as accurate as the original IsoRank. In fact, for functional enrichment-based measures of global network alignment quality we find our approximation performs better than exact IsoRank, doubtless because it is more robust to the noise of missing or incorrect edges. It also performs competitively against two more recent global network alignment algorithms.

In this paper, we present a pressure-robust enriched Galerkin (EG) scheme for solving the Stokes equations, which is an enhanced version of the EG scheme for the Stokes problem proposed in Yi et al. (2022). The pressure-robustness is achieved by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. An a priori error analysis proves that the velocity error is independent of the pressure and viscosity. We also propose and analyze a perturbed version of our pressure-robust EG method that allows for the elimination of the degrees of freedom corresponding to the discontinuous component of the velocity vector via static condensation. The resulting method can be viewed as a stabilized H1-conforming P1−P0 method. Further, we consider efficient block preconditioners whose performances are independent of the viscosity. The theoretical results are confirmed through various numerical experiments in two and three dimensions.

Machine learning techniques have recently been of great interest for solving differential equations. Training these models is classically a data-fitting task, but knowledge of the expression of the differential equation can be used to supplement the training objective, leading to the development of physics-informed scientific machine learning. In this article, we focus on one class of models called nonlinear vector autoregression (NVAR) to solve ordinary differential equations (ODEs). Motivated by connections to numerical integration and physics-informed neural networks, we explicitly derive the physics-informed NVAR (piNVAR) which enforces the right-hand side of the underlying differential equation regardless of NVAR construction. Because NVAR and piNVAR completely share their learned parameters, we propose an augmented procedure to jointly train the two models. Then, using both data-driven and ODE-driven metrics, we evaluate the ability of the piNVAR model to predict solutions to various ODE systems, such as the undamped spring, a Lotka-Volterra predator-prey nonlinear model, and the chaotic Lorenz system.

In this work, a local Fourier analysis is presented to study the convergence of multigrid methods based on additive Schwarz smoothers. This analysis is presented as a general framework which allows us to study these smoothers for any type of discretization and problem. The presented framework is crucial in practice since it allows one to know a priori the answer to questions such as what is the size of the patch to use within these relaxations, the size of the overlap, or even the optimal values for the weights involved in the smoother. Results are shown for a class of additive and restricted additive Schwarz relaxations used within a multigrid framework applied to high-order finite-element discretizations and saddle point problems, which are two of the contexts in which these types of relaxations are widely used.

Fractional vector calculus is the building block of the fractional partial differential equations that model non-local or long-range phenomena, e.g., anomalous diffusion, fractional electromagnetism, and fractional advection-dispersion. In this work, we reformulate a type of fractional vector calculus that uses Caputo fractional partial derivatives and discretize this reformulation using discrete exterior calculus on a cubical complex in the structure-preserving way, meaning that the continuous-level properties curlαgradα=0 and divαcurlα=0 hold exactly on the discrete level. We discuss important properties of our fractional discrete exterior derivatives and verify their second-order convergence in the root mean square error numerically. Our proposed discretization has the potential to provide accurate and stable numerical solutions to fractional partial differential equations and exactly preserve fundamental physics laws on the discrete level regardless of the mesh size.