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Modified Patankar-Runge-Kutta (MPRK) methods preserve the positivity as well as conservativity of a production-destruction system (PDS) of ordinary differential equations for all time step sizes. As a result, higher order MPRK schemes do not belong to the class of general linear methods, i.e. the iterates are generated by a nonlinear map $\mathbf g$ even when the PDS is linear. Moreover, due to the conservativity of the method, the map $\mathbf g$ possesses non-hyperbolic fixed points. Recently, a new theorem for the investigation of stability properties of non-hyperbolic fixed points of a nonlinear iteration map was developed. We apply this theorem to understand the stability properties of a family of second order MPRK methods when applied to a nonlinear PDS of ordinary differential equations. It is shown that the fixed points are stable for all time step sizes and members of the MPRK family. Finally, experiments are presented to numerically support the theoretical claims.

The paper proposes a decoupled numerical scheme of the time-dependent Ginzburg-Landau equations under temporal gauge. For the order parameter and the magnetic potential, the discrete scheme adopts the second type Ned${\rm \acute{e}}$lec element and the linear element for spatial discretization, respectively, and a fully linearized backward Euler method and the first order exponential time differencing method for time discretization, respectively. The maximum bound principle of the order parameter and the energy dissipation law in the discrete sense are proved for this finite element-based scheme. This allows the application of the adaptive time stepping method which can significantly speed up long-time simulations compared to existing numerical schemes, especially for superconductors with complicated shapes. The error estimate is rigorously established in the fully discrete sense. Numerical examples verify the theoretical results of the proposed scheme and demonstrate the vortex motions of superconductors in an external magnetic field.

In this paper, we investigate a general class of stochastic gradient descent (SGD) algorithms, called conditioned SGD, based on a preconditioning of the gradient direction. Using a discrete-time approach with martingale tools, we establish the weak convergence of the rescaled sequence of iterates for a broad class of conditioning matrices including stochastic first-order and second-order methods. Almost sure convergence results, which may be of independent interest, are also presented. When the conditioning matrix is an estimate of the inverse Hessian, the algorithm is proved to be asymptotically optimal. For the sake of completeness, we provide a practical procedure to achieve this minimum variance.

Two-stage randomized experiments are becoming an increasingly popular experimental design for causal inference when the outcome of one unit may be affected by the treatment assignments of other units in the same cluster. In this paper, we provide a methodological framework for general tools of statistical inference and power analysis for two-stage randomized experiments. Under the randomization-based framework, we consider the estimation of a new direct effect of interest as well as the average direct and spillover effects studied in the literature. We provide unbiased estimators of these causal quantities and their conservative variance estimators in a general setting. Using these results, we then develop hypothesis testing procedures and derive sample size formulas. We theoretically compare the two-stage randomized design with the completely randomized and cluster randomized designs, which represent two limiting designs. Finally, we conduct simulation studies to evaluate the empirical performance of our sample size formulas. For empirical illustration, the proposed methodology is applied to the randomized evaluation of the Indian national health insurance program. An open-source software package is available for implementing the proposed methodology.

Neural ordinary differential equations (Neural ODEs) model continuous time dynamics as differential equations parametrized with neural networks. Thanks to their modeling flexibility, they have been adopted for multiple tasks where the continuous time nature of the process is specially relevant, as in system identification and time series analysis. When applied in a control setting, it is possible to adapt their use to approximate optimal nonlinear feedback policies. This formulation follows the same approach as policy gradients in reinforcement learning, covering the case where the environment consists of known deterministic dynamics given by a system of differential equations. The white box nature of the model specification allows the direct calculation of policy gradients through sensitivity analysis, avoiding the inexact and inefficient gradient estimation through sampling. In this work we propose the use of a neural control policy posed as a Neural ODE to solve general nonlinear optimal control problems while satisfying both state and control constraints, which are crucial for real world scenarios. Since the state feedback policy partially modifies the model dynamics, the whole space phase of the system is reshaped upon the optimization. This approach is a sensible approximation to the historically intractable closed loop solution of nonlinear control problems that efficiently exploits the availability of a dynamical system model.

This paper is concerned with a blood flow problem coupled with a slow plaque growth at the artery wall. In the model, the micro (fast) system is the Navier-Stokes equation with a periodically applied force and the macro (slow) system is a fractional reaction equation, which is used to describe the plaque growth with memory effect. We construct an auxiliary temporal periodic problem and an effective time-average equation to approximate the original problem and analyze the approximation error of the corresponding linearized PDE (Stokes) system, where the simple front-tracking technique is used to update the slow moving boundary. An effective multiscale method is then designed based on the approximate problem and the front tracking framework. We also present a temporal finite difference scheme with a spatial continuous finite element method and analyze its temporal discrete error. Furthermore, a fast iterative procedure is designed to find the initial value of the temporal periodic problem and its convergence is analyzed as well. Our designed front-tracking framework and the iterative procedure for solving the temporal periodic problem make it easy to implement the multiscale method on existing PDE solving software. The numerical method is implemented by a combination of the finite element platform COMSOL Multiphysics and the mainstream software MATLAB, which significantly reduce the programming effort and easily handle the fluid-structure interaction, especially moving boundaries with more complex geometries. We present some numerical examples of ODEs and 2-D Navier-Stokes system to demonstrate the effectiveness of the multiscale method. Finally, we have a numerical experiment on the plaque growth problem and discuss the physical implication of the fractional order parameter.

Learning policies via preference-based reward learning is an increasingly popular method for customizing agent behavior, but has been shown anecdotally to be prone to spurious correlations and reward hacking behaviors. While much prior work focuses on causal confusion in reinforcement learning and behavioral cloning, we aim to study it in the context of reward learning. To study causal confusion, we perform a series of sensitivity and ablation analyses on three benchmark domains where rewards learned from preferences achieve minimal test error but fail to generalize to out-of-distribution states -- resulting in poor policy performance when optimized. We find that the presence of non-causal distractor features, noise in the stated preferences, partial state observability, and larger model capacity can all exacerbate causal confusion. We also identify a set of methods with which to interpret causally confused learned rewards: we observe that optimizing causally confused rewards drives the policy off the reward's training distribution, resulting in high predicted (learned) rewards but low true rewards. These findings illuminate the susceptibility of reward learning to causal confusion, especially in high-dimensional environments -- failure to consider even one of many factors (data coverage, state definition, etc.) can quickly result in unexpected, undesirable behavior.

This paper presents a novel physics-infused reduced-order modeling (PIROM) methodology for efficient and accurate modeling of non-linear dynamical systems. The PIROM consists of a physics-based analytical component that represents the known physical processes, and a data-driven dynamical component that represents the unknown physical processes. The PIROM is applied to the aerothermal load modeling for hypersonic aerothermoelastic (ATE) analysis and is found to accelerate the ATE simulations by two-three orders of magnitude while maintaining an accuracy comparable to high-fidelity solutions based on computational fluid dynamics (CFD). Moreover, the PIROM-based solver is benchmarked against the conventional POD-kriging surrogate model, and is found to significantly outperform the accuracy, generalizability and sampling efficiency of the latter in a wide range of operating conditions and in the presence of complex structural boundary conditions. Finally, the PIROM-based ATE solver is demonstrated by a parametric study on the effects of boundary conditions and rib-supports on the ATE response of a compliant and heat-conducting panel structure. The results not only reveal the dramatic snap-through behavior with respect to spring constraints of boundary conditions, but also demonstrates the potential of PIROM to facilitate the rapid and accurate design and optimization of multi-disciplinary systems such as hypersonic structures.

In this work, a fully adaptive finite element algorithm for symmetric second-order elliptic diffusion problems with inexact solver is developed. The discrete systems are treated by a local higher-order geometric multigrid method extending the approach of [Mira\c{c}i, Pape\v{z}, Vohral\'{i}k, SIAM J. Sci. Comput. (2021)]. We show that the iterative solver contracts the algebraic error robustly with respect to the polynomial degree $p \ge 1$ and the (local) mesh size $h$. We further prove that the built-in algebraic error estimator is $h$- and $p$-robustly equivalent to the algebraic error. The proofs rely on suitably chosen robust stable decompositions and a strengthened Cauchy-Schwarz inequality on bisection-generated meshes. Together, this yields that the proposed adaptive algorithm has optimal computational cost. Numerical experiments confirm the theoretical findings.

We introduce a Fourier-based fast algorithm for Gaussian process regression. It approximates a translationally-invariant covariance kernel by complex exponentials on an equispaced Cartesian frequency grid of $M$ nodes. This results in a weight-space $M\times M$ system matrix with Toeplitz structure, which can thus be applied to a vector in ${\mathcal O}(M \log{M})$ operations via the fast Fourier transform (FFT), independent of the number of data points $N$. The linear system can be set up in ${\mathcal O}(N + M \log{M})$ operations using nonuniform FFTs. This enables efficient massive-scale regression via an iterative solver, even for kernels with fat-tailed spectral densities (large $M$). We include a rigorous error analysis of the kernel approximation, the resulting accuracy (relative to "exact" GP regression), and the condition number. Numerical experiments for squared-exponential and Mat\'ern kernels in one, two and three dimensions often show 1-2 orders of magnitude acceleration over state-of-the-art rank-structured solvers at comparable accuracy. Our method allows 2D Mat\'ern-${\small \frac{3}{2}}$ regression from $N=10^9$ data points to be performed in 2 minutes on a standard desktop, with posterior mean accuracy $10^{-3}$. This opens up spatial statistics applications 100 times larger than previously possible.