This paper investigates, a new class of fractional order Runge-Kutta (FORK) methods for numerical approximation to the solution of fractional differential equations (FDEs). By using the Caputo generalized Taylor formula and the total differential for Caputo fractional derivative, we construct explicit and implicit FORK methods, as the well-known Runge-Kutta schemes for ordinary differential equations. In the proposed method, due to the dependence of fractional derivatives to a fixed base point $t_0,$ we had to modify the right-hand side of the given equation in all steps of the FORK methods. Some coefficients for explicit and implicit FORK schemes are presented. The convergence analysis of the proposed method is also discussed. Numerical experiments clarify the effectiveness and robustness of the method.
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Charge dynamics play essential role in many practical applications such as semiconductors, electrochemical devices and transmembrane ion channels. A Maxwell-Amp\`{e}re Nernst-Planck (MANP) model that describes charge dynamics via concentrations and the electric displacement is able to take effects beyond mean-field approximations into account. To obtain physically faithful numerical solutions, we develop a structure-preserving numerical method for the MANP model whose solution has several physical properties of importance. By the Slotboom transform with entropic-mean approximations, a positivity preserving scheme with Scharfetter-Gummel fluxes is derived for the generalized Nernst-Planck equations. To deal with the curl-free constraint, the dielectric displacement from the Maxwell-Amp\`{e}re equation is further updated with a local relaxation algorithm of linear computational complexity. We prove that the proposed numerical method unconditionally preserves the mass conservation and the solution positivity at the discrete level, and satisfies the discrete energy dissipation law with a time-step restriction. Numerical experiments verify that our numerical method has expected accuracy and structure-preserving properties. Applications to ion transport with large convection, arising from boundary-layer electric field and Born solvation interactions, further demonstrate that the MANP formulation with the proposed numerical scheme has attractive performance and can effectively describe charge dynamics with large convection of high numerical cell P\'{e}clet numbers.
Given ample experimental data from a system governed by differential equations, it is possible to use deep learning techniques to construct the underlying differential operators. In this work we perform symbolic discovery of differential operators in a situation where there is sparse experimental data. This small data regime in machine learning can be made tractable by providing our algorithms with prior information about the underlying dynamics. Physics Informed Neural Networks (PINNs) have been very successful in this regime (reconstructing entire ODE solutions using only a single point or entire PDE solutions with very few measurements of the initial condition). We modify the PINN approach by adding a neural network that learns a representation of unknown hidden terms in the differential equation. The algorithm yields both a surrogate solution to the differential equation and a black-box representation of the hidden terms. These hidden term neural networks can then be converted into symbolic equations using symbolic regression techniques like AI Feynman. In order to achieve convergence of these neural networks, we provide our algorithms with (noisy) measurements of both the initial condition as well as (synthetic) experimental data obtained at later times. We demonstrate strong performance of this approach even when provided with very few measurements of noisy data in both the ODE and PDE regime.
A classical tool for approximating integrals is the Laplace method. The first-order, as well as the higher-order Laplace formula is most often written in coordinates without any geometrical interpretation. In this article, motivated by a situation arising, among others, in optimal transport, we give a geometric formulation of the first-order term of the Laplace method. The central tool is the Kim-McCann Riemannian metric which was introduced in the field of optimal transportation. Our main result expresses the first-order term with standard geometric objects such as volume forms, Laplacians, covariant derivatives and scalar curvatures of two different metrics arising naturally in the Kim-McCann framework. Passing by, we give an explicitly quantified version of the Laplace formula, as well as examples of applications.
An implicit variable-step BDF2 scheme is established for solving the space fractional Cahn-Hilliard equation, involving the fractional Laplacian, derived from a gradient flow in the negative order Sobolev space $H^{-\alpha}$, $\alpha\in(0,1)$. The Fourier pseudo-spectral method is applied for the spatial approximation. The proposed scheme inherits the energy dissipation law in the form of the modified discrete energy under the sufficient restriction of the time-step ratios. The convergence of the fully discrete scheme is rigorously provided utilizing the newly proved discrete embedding type convolution inequality dealing with the fractional Laplacian. Besides, the mass conservation and the unique solvability are also theoretically guaranteed. Numerical experiments are carried out to show the accuracy and the energy dissipation both for various interface widths. In particular, the multiple-time-scale evolution of the solution is captured by an adaptive time-stepping strategy in the short-to-long time simulation.
Existing techniques for training language models can be misaligned with the truth: if we train models with imitation learning, they may reproduce errors that humans make; if we train them to generate text that humans rate highly, they may output errors that human evaluators can't detect. We propose circumventing this issue by directly finding latent knowledge inside the internal activations of a language model in a purely unsupervised way. Specifically, we introduce a method for accurately answering yes-no questions given only unlabeled model activations. It works by finding a direction in activation space that satisfies logical consistency properties, such as that a statement and its negation have opposite truth values. We show that despite using no supervision and no model outputs, our method can recover diverse knowledge represented in large language models: across 6 models and 10 question-answering datasets, it outperforms zero-shot accuracy by 4\% on average. We also find that it cuts prompt sensitivity in half and continues to maintain high accuracy even when models are prompted to generate incorrect answers. Our results provide an initial step toward discovering what language models know, distinct from what they say, even when we don't have access to explicit ground truth labels.
We prove $hp$-optimal error estimates for interior penalty discontinuous Galerkin methods (IPDG) for the biharmonic problem with homogeneous essential boundary conditions. We consider tensor product-type meshes in two and three dimensions, and triangular meshes in two dimensions. An essential ingredient in the analysis is the construction of a global $H^2$ piecewise polynomial approximants with $hp$-optimal approximation properties over the given meshes. The $hp$-optimality is also discussed for $\mathcal C^0$-IPDG in two and three dimensions, and the stream formulation of the Stokes problem in two dimensions. Numerical experiments validate the theoretical predictions and reveal that $p$-suboptimality occurs in presence of singular essential boundary conditions.
This paper develops fast and accurate linear finite element method and fourth-order compact difference method combined with matrix transfer technique to solve high dimensional time-space fractional diffusion problem with spectral fractional Laplacian in space. In addition, a fast time stepping $L1$ scheme is used for time discretization. We can exactly evaluate fractional power of matrix in the proposed schemes, and perform matrix-vector multiplication by directly using a discrete sine transform and its inverse transform, which doesn't need to resort to any iteration method and can significantly reduce computation cost and memory. Further, we address the convergence analyses of full discrete scheme based on two types of spatial numerical methods. Finally, ample numerical examples are delivered to illustrate our theoretical analyses and the efficiency of the suggested schemes.
Deep semantic matching aims to discriminate the relationship between documents based on deep neural networks. In recent years, it becomes increasingly popular to organize documents with a graph structure, then leverage both the intrinsic document features and the extrinsic neighbor features to derive discrimination. Most of the existing works mainly care about how to utilize the presented neighbors, whereas limited effort is made to filter appropriate neighbors. We argue that the neighbor features could be highly noisy and partially useful. Thus, a lack of effective neighbor selection will not only incur a great deal of unnecessary computation cost, but also restrict the matching accuracy severely. In this work, we propose a novel framework, Cascaded Deep Semantic Matching (CDSM), for accurate and efficient semantic matching on textual graphs. CDSM is highlighted for its two-stage workflow. In the first stage, a lightweight CNN-based ad-hod neighbor selector is deployed to filter useful neighbors for the matching task with a small computation cost. We design both one-step and multi-step selection methods. In the second stage, a high-capacity graph-based matching network is employed to compute fine-grained relevance scores based on the well-selected neighbors. It is worth noting that CDSM is a generic framework which accommodates most of the mainstream graph-based semantic matching networks. The major challenge is how the selector can learn to discriminate the neighbors usefulness which has no explicit labels. To cope with this problem, we design a weak-supervision strategy for optimization, where we train the graph-based matching network at first and then the ad-hoc neighbor selector is learned on top of the annotations from the matching network.
Physical laws governing population dynamics are generally expressed as differential equations. Research in recent decades has incorporated fractional-order (non-integer) derivatives into differential models of natural phenomena, such as reaction-diffusion systems. In this paper, we develop a method to numerically solve a multi-component and multi-dimensional space-fractional system. For space discretization, we apply a Fourier spectral method that is suited for multidimensional PDE systems. Efficient approximation of time-stepping is accomplished with a locally one dimensional exponential time differencing approach. We show the effect of different fractional parameters on growth models and consider the convergence, stability, and uniqueness of solutions, as well as the biological interpretation of parameters and boundary conditions.
In order to overcome the expressive limitations of graph neural networks (GNNs), we propose the first method that exploits vector flows over graphs to develop globally consistent directional and asymmetric aggregation functions. We show that our directional graph networks (DGNs) generalize convolutional neural networks (CNNs) when applied on a grid. Whereas recent theoretical works focus on understanding local neighbourhoods, local structures and local isomorphism with no global information flow, our novel theoretical framework allows directional convolutional kernels in any graph. First, by defining a vector field in the graph, we develop a method of applying directional derivatives and smoothing by projecting node-specific messages into the field. Then we propose the use of the Laplacian eigenvectors as such vector field, and we show that the method generalizes CNNs on an n-dimensional grid, and is provably more discriminative than standard GNNs regarding the Weisfeiler-Lehman 1-WL test. Finally, we bring the power of CNN data augmentation to graphs by providing a means of doing reflection, rotation and distortion on the underlying directional field. We evaluate our method on different standard benchmarks and see a relative error reduction of 8\% on the CIFAR10 graph dataset and 11% to 32% on the molecular ZINC dataset. An important outcome of this work is that it enables to translate any physical or biological problems with intrinsic directional axes into a graph network formalism with an embedded directional field.
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