Approximation Of Differential Equations Research Articles

Gradient-based adaptive neural network technique for two-dimensional local fractional elliptic PDEs

This paper introduces gradient-based adaptive neural networks to solve local fractional elliptic partial differential equations. The impact of physics-informed neural networks helps to approximate elliptic partial differential equations governed by the physical process. The proposed technique employs learning the behaviour of complex systems based on input-output data, and automatic differentiation ensures accurate computation of gradient. The method computes the singularity-embedded local fractional partial derivative model on a Hausdorff metric, which otherwise halts the computation by available approximating numerical methods. This is possible because the new network is capable of updating the weight associated with loss terms depending on the solution domain and requirement of solution behaviour. The semi-positive definite character of the neural tangent kernel achieves the convergence of gradient-based adaptive neural networks. The importance of hyperparameters, namely the number of neurons and the learning rate, is shown by considering a stationary anomalous diffusion-convection model on a rectangular domain. The proposed method showcases the network’s ability to approximate solutions of various local fractional elliptic partial differential equations with varying fractal parameters.

On the complexity of strong approximation of stochastic differential equations with a non-Lipschitz drift coefficient

We survey recent developments in the field of complexity of pathwise approximation in p-th mean of the solution of a stochastic differential equation at the final time based on finitely many evaluations of the driving Brownian motion. First, we briefly review the case of equations with globally Lipschitz continuous coefficients, for which an error rate of at least 1/2 in terms of the number of evaluations of the driving Brownian motion is always guaranteed by using the equidistant Euler-Maruyama scheme. Then we illustrate that giving up the global Lipschitz continuity of the coefficients may lead to a non-polynomial decay of the error for the Euler-Maruyama scheme or even to an arbitrary slow decay of the smallest possible error that can be achieved on the basis of finitely many evaluations of the driving Brownian motion. Finally, we turn to recent positive results for equations with a drift coefficient that is not globally Lipschitz continuous. Here we focus on scalar equations with a Lipschitz continuous diffusion coefficient and a drift coefficient that satisfies piecewise smoothness assumptions or has fractional Sobolev regularity and we present corresponding complexity results.

Unveiling the Power of Implicit Six-Point Block Scheme: Advancing numerical approximation of two-dimensional PDEs in physical systems.

In the era of computational advancements, harnessing computer algorithms for approximating solutions to differential equations has become indispensable for its unparalleled productivity. The numerical approximation of partial differential equation (PDE) models holds crucial significance in modelling physical systems, driving the necessity for robust methodologies. In this article, we introduce the Implicit Six-Point Block Scheme (ISBS), employing a collocation approach for second-order numerical approximations of ordinary differential equations (ODEs) derived from one or two-dimensional physical systems. The methodology involves transforming the governing PDEs into a fully-fledged system of algebraic ordinary differential equations by employing ISBS to replace spatial derivatives while utilizing a central difference scheme for temporal or y-derivatives. In this report, the convergence properties of ISBS, aligning with the principles of multi-step methods, are rigorously analyzed. The numerical results obtained through ISBS demonstrate excellent agreement with theoretical solutions. Additionally, we compute absolute errors across various problem instances, showcasing the robustness and efficacy of ISBS in practical applications. Furthermore, we present a comprehensive comparative analysis with existing methodologies from recent literature, highlighting the superior performance of ISBS. Our findings are substantiated through illustrative tables and figures, underscoring the transformative potential of ISBS in advancing the numerical approximation of two-dimensional PDEs in physical systems.

A certified wavelet-based physics-informed neural network for the solution of parameterized partial differential equations

Abstract Physics Informed Neural Networks (PINNs) have frequently been used for the numerical approximation of Partial Differential Equations (PDEs). The goal of this paper is to construct PINNs along with a computable upper bound of the error, which is particularly relevant for model reduction of Parameterized PDEs (PPDEs). To this end, we suggest to use a weighted sum of expansion coefficients of the residual in terms of an adaptive wavelet expansion both for the loss function and an error bound. This approach is shown here for elliptic PPDEs using both the standard variational and an optimally stable ultra-weak formulation. Numerical examples show a very good quantitative effectivity of the wavelet-based error bound.

Galerkin approximation for multi-term time-fractional differential equations

Fractional differential equations (FDEs) are utilized as a precise model for describing a wide range of biological and physical processes, benefiting from the inherent symmetry feature in natural phenomena. The introduction of multi-term time-fractional order equations serves to tackle intricate phenomena spanning various physical domains. This study solves multi-term time-fractional DEs using the quadratic B-spline Galerkin method. In Galerkin approach, the quadratic B-spline is utilized as both a test and a basis function. The fractional portion of time is evaluated using the Caputo definition. In addition, the Gauss quadrature formula is used to evaluate the integration of complex function which enhance the accuracy. Lax-Richtmyer's stability criterion is employed to evaluate the stability of the proposed scheme. The E2 and E∞ norms are computed and displayed in the tables to illustrate the robustness and efficiency of the method. We compute simulation for various nodal points as well as various mesh sizes to analyze the solution behavior. The obtained results are compared with the exact and available results, indicating that the proposed scheme is more efficient. The graphical solution of each example is also demonstrated which verifies that exact and approximate solutions graphs closed with each other.

High-order bounds-satisfying approximation of partial differential equations via finite element variational inequalities

High-order bounds-satisfying approximation of partial differential equations via finite element variational inequalities

Spatially dependent node regularity in meshless approximation of partial differential equations

In this paper, we address a way to reduce the total computational cost of meshless approximation by reducing the required stencil size through spatially varying computational node regularity. Rather than covering the entire domain with scattered nodes, only regions with geometric details are covered with scattered nodes, while the rest of the domain is discretized with regular nodes. A simpler approximation can be used in regions covered by regular nodes, effectively reducing the required stencil size and computational cost compared to the approximation on scattered nodes where a set of polyharmonic splines is added to ensure convergent behaviour.This paper is an extended version of conference paper entitled “Spatially-varying meshless approximation method for enhanced computational efficiency” (Jančič et al., 2023) presented at “International Conference on Computational Science (ICCS) 2023”. The paper is extended with discussion on development and implementation of a hybrid regular-scattered node positioning algorithm (HyNP). The performance of the proposed HyNP algorithm is analysed in terms of separation distance and maximal empty sphere radius. Furthermore, it is demonstrated that HyNP nodes can be used for solving problems from fluid flow and linear elasticity, both in 2D and 3D, using meshless methods.The extension also provides additional analyses of computational efficiency and accuracy of the numerical solution obtained on the spatially-variable regularity of discretization nodes. In particular, different levels of refinement aggressiveness and scattered layer widths are considered to exploit the computational efficiency gains offered by such solution procedure.

Approximate Noether theorem and its inverse for nonlinear dynamical systems with approximate nonstandard Lagrangian

The approximate Noether theorem and its inverse theorem for the nonlinear dynamical systems with approximate exponential Lagrangian and approximate power-law Lagrangian are investigated. For each case, the approximate differential equations of motion for the nonlinear dynamical systems with approximate nonstandard Lagrangian are established, the generalized Noether identities are given. The relationship between the approximate Noether symmetries and approximate conserved quantities for the system with approximate nonstandard Lagrangian are established, and the approximate Noether theorems and their inverse theorems are obtained. Two examples are given to illustrate the application of the results.

Optimal transport with nonlinear mobilities: A deterministic particle approximation result

Abstract We study the discretisation of generalised Wasserstein distances with nonlinear mobilities on the real line via suitable discrete metrics on the cone of N ordered particles, a setting which naturally appears in the framework of deterministic particle approximation of partial differential equations. In particular, we provide a Γ-convergence result for the associated discrete metrics as N → ∞ {N\to\infty} to the continuous one and discuss applications to the approximation of one-dimensional conservation laws (of gradient flow type) via the so-called generalised minimising movements, proving a convergence result of the schemes at any given discrete time step τ > 0 {\tau>0} . This the first work of a series aimed at sheding new lights on the interplay between generalised gradient-flow structures, conservation laws, and Wasserstein distances with nonlinear mobilities.

Wong–Zakai approximation of a stochastic partial differential equation with multiplicative noise

In this article, we derive the convergence rate for the Wong–Zakai equation of some approximation of stochastic evolution equations with multiplicative noise. To be more precise, the diffusion coefficient in front of the noise is the multiplication operator, and, is therefore not bounded, a situation not treated in the literature. Since our motivation comes from problems in numerical ling, we consider a finite, high-dimensional problem approximating a stochastic evolution equation on a random time grid. By imposing suitable stability conditions on the drift term and the time grid, we achieve a convergence rate in the mean square of order min { 1 − δ , 2 − 2 γ } , for some 0 < δ < 1 and 0 < γ < 1 / 2 .

Multilayered neural network for power series‐based approximation of fractional delay differential equations

This paper trains a multilayered neural network (MLNN) for solving fractional delay differential equations (FDDEs), including nonlinear and singular types. The proposed methodology involves replacing the unknown functions in the equations with a truncated power series expansion. Subsequently, a collection of algebraic equations is solved utilizing an iterative minimization technique that leverages the capabilities of the MLNN architecture. The outcomes demonstrate that the MLNN architecture provides the required accuracy and strong stability compared to several numerical methods.

Near-integrable dynamics of the Fermi-Pasta-Ulam-Tsingou problem.

It is well known that the classic Fermi-Pasta-Ulam-Tsingou (FPUT) study of a chain of nonlinear oscillators is closely related to a number of completely integrable systems, including the Toda lattice. Here, we present a method that captures the departure of nonintegrable FPUT dynamics from those of a nearby integrable Toda lattice. Using initial long-wave data, we find that the former depart rather sharply from the latter near the predicted shock time of an asymptotic partial differential equation approximation, at which point energy cascades into higher lattice modes. Our method provides an appropriate frame of reference for one to distinguish the short-term dynamics of the two systems, whose macroscopic trajectories diverge noticeably only on a much longer timescale, when the FPUT dynamics thermalize.

Exponential Euler method for stiff stochastic differential equations with additive fractional Brownian noise

We discuss a system of stochastic differential equations with a stiff linear term and additive noise driven by fractional Brownian motions (fBms) with Hurst parameter H > 1 2 , which arise e.g. from spatial approximations of stochastic partial differential equations. For their numerical approximation, we present an exponential Euler scheme and show that it converges in the strong sense with an exact rate close to the Hurst parameter H. Further, based on E. Buckwar et al. [The numerical stability of stochastic ordinary differential equations with additive noise, Stoch. Dyn. 11 (2011), pp. 265–281], we conclude the existence of a unique stationary solution of the exponential Euler scheme that is pathwise asymptotically stable.

Learning the intrinsic dynamics of spatio-temporal processes through Latent Dynamics Networks

Predicting the evolution of systems with spatio-temporal dynamics in response to external stimuli is essential for scientific progress. Traditional equations-based approaches leverage first principles through the numerical approximation of differential equations, thus demanding extensive computational resources. In contrast, data-driven approaches leverage deep learning algorithms to describe system evolution in low-dimensional spaces. We introduce an architecture, termed Latent Dynamics Network, capable of uncovering low-dimensional intrinsic dynamics in potentially non-Markovian systems. Latent Dynamics Networks automatically discover a low-dimensional manifold while learning the system dynamics, eliminating the need for training an auto-encoder and avoiding operations in the high-dimensional space. They predict the evolution, even in time-extrapolation scenarios, of space-dependent fields without relying on predetermined grids, thus enabling weight-sharing across query-points. Lightweight and easy-to-train, Latent Dynamics Networks demonstrate superior accuracy (normalized error 5 times smaller) in highly-nonlinear problems with significantly fewer trainable parameters (more than 10 times fewer) compared to state-of-the-art methods.

Fractional Trigonometric Function Approximations in a Regular System

Introduction: Linear fractional differential equations of the commensurable order, specifically those related to fractional trigonometry, pose challenges in terms of analytical solutions. background: This study presents an approach to the solution of fractional differential equations with commensurate order, specifically those associated with fractional trigonometry. Methods: The purpose of this article is to present a novel approximate analytical technique to solve linear fractional differential equations of the commensurable order, which are related to fractional trigonometry. This method is used to obtain approximate solutions by forming linear combinations of appropriate fractional basic functions, for which Laplace transforms are irrational functions. This approximation technique enables us to represent these functions using timeinvariant linear system models. Also, the implementation of analog circuits of these basic fractional order systems can be obtained using approximation of rational functions. Results: The research demonstrates the efficacy and precision of this method through illustrative examples, showcasing its effectiveness in solving linear fractional systems. Conclusion: The newly introduced approximate analytical method presents a promising approach to solving linear fractional differential equations of the commensurable order, providing accurate solutions and possibly offering applications in various fields.

Numerical Approximation for a Stochastic Fractional Differential Equation Driven by Integrated Multiplicative Noise

We consider a numerical approximation for stochastic fractional differential equations driven by integrated multiplicative noise. The fractional derivative is in the Caputo sense with the fractional order α∈(0,1), and the non-linear terms satisfy the global Lipschitz conditions. We first approximate the noise with the piecewise constant function to obtain the regularized stochastic fractional differential equation. By applying Minkowski’s inequality for double integrals, we establish that the error between the exact solution and the solution of the regularized problem has an order of O(Δtα) in the mean square norm, where Δt denotes the step size. To validate our theoretical conclusions, numerical examples are presented, demonstrating the consistency of the numerical results with the established theory.

L2-convergence of Yosida approximation for semi-linear backward stochastic differential equation with jumps in infinite dimension

PurposeThe main motivation of this paper is to present the Yosida approximation of a semi-linear backward stochastic differential equation in infinite dimension. Under suitable assumption and condition, an L2-convergence rate is established.Design/methodology/approachThe authors establish a result concerning the L2-convergence rate of the solution of backward stochastic differential equation with jumps with respect to the Yosida approximation.Findings The authors carry out a convergence rate of Yosida approximation to the semi-linear backward stochastic differential equation in infinite dimension.Originality/valueIn this paper, the authors present the Yosida approximation of a semi-linear backward stochastic differential equation in infinite dimension. Under suitable assumption and condition, an L2-convergence rate is established.

Effective reduced models from delay differential equations: Bifurcations, tipping solution paths, and ENSO variability

Conceptual delay models have played a key role in the analysis and understanding of El Niño-Southern Oscillation (ENSO) variability. Based on such delay models, we propose in this work a novel scenario for the fabric of ENSO variability resulting from the subtle interplay between stochastic disturbances and nonlinear invariant sets emerging from bifurcations of the unperturbed dynamics.To identify these invariant sets we adopt an approach combining Galerkin–Koornwinder (GK) approximations of delay differential equations and center-unstable manifold reduction techniques. In that respect, GK approximation formulas are reviewed and synthesized, as well as analytic approximation formulas of center-unstable manifolds. The reduced systems derived thereof enable us to conduct a thorough analysis of the bifurcations arising in a standard delay model of ENSO. We identify thereby a saddle–node bifurcation of periodic orbits co-existing with a subcritical Hopf bifurcation, and a homoclinic bifurcation for this model. We show furthermore that the computation of unstable periodic orbits (UPOs) unfolding through these bifurcations is considerably simplified from the reduced systems.These dynamical insights enable us in turn to design a stochastic model whose solutions – as the delay parameter drifts slowly through its critical values – produce a wealth of temporal patterns resembling ENSO events and exhibiting also decadal variability. Our analysis dissects the origin of this variability and shows how it is tied to certain transition paths between invariant sets of the unperturbed dynamics (for ENSO’s interannual variability) or simply due to the presence of UPOs close to the homoclinic orbit (for decadal variability). In short, this study points out the role of solution paths evolving through tipping “points” beyond equilibria, as possible mechanisms organizing the variability of certain climate phenomena.

Analytic regularity and solution approximation for a semilinear elliptic partial differential equation in a polygon

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Numerical approximations of stochastic delay differential equations with delayed impulses

This paper focuses on approximating stochastic delay differential equations with delayed impulses using Euler-Maruyama-type approximations. One key difference from previous literature is that the impulsive perturbations considered in this paper are past-dependent. Additionally, both the time delays in the stochastic delay differential equations and in the impulsive functions are functions of time. We establish the mean square convergence of the Euler-Maruyama approximations under a local Lipschitz condition and a linear growth condition. Furthermore, we determine the order of convergence under a global Lipschitz condition and provide an illustrative example.