arXiv subjects

Anshima Singh

Publications and source records attributed to Anshima Singh.

5 recordsLinked to original sources

arXiv2026

DAS-PINNs for high-dimensional partial differential equations: extending deep adaptive sampling to spacetime domains

Time-dependent high-dimensional partial differential equations (PDEs) with spatially localised and dynamically evolving solutions pose a fundamental challenge for physics-informed neural networks (PINNs), as uniform collocation sampling becomes increasingly ineffective in high-dimensional spatiotemporal domains. In this work, a deep adaptive sampling framework for PINNs is extended to the time-dependent setting by treating space and time as a unified domain without any explicit time marching. A normalising flow neural network model effectively learns the distribution induced by the PDE residual and generates new collocation points concentrated in regions where the solution is most difficult to learn. Unlike conventional adaptive strategies that require explicit time stepping or moving meshes, high-residual regions are automatically identified and tracked across both space and time, driven purely by the PDE residual distribution. The effectiveness of the proposed strategy is assessed on a range of benchmark problems, from sharp and moving features in two spatial dimensions to localised structures in up to eight spatial dimensions.

math.NA↗

arXiv2025

Adaptive Sampling for Hydrodynamic Stability

An adaptive sampling approach for efficient detection of bifurcation boundaries in parametrized fluid flow problems is presented herein. The study extends the machine-learning approach of Silvester~(J. Comput. Phys., 553 (2026), 114743), where a classifier network was trained on preselected simulation data to identify bifurcated and nonbifurcated flow regimes. In contrast, the proposed methodology introduces adaptivity through a flow-based deep generative model that automatically refines the sampling of the parameter space. The strategy has two components: a classifier network maps the flow parameters to a bifurcation probability, and a probability density estimation technique (KRnet) for the generation of new samples at each adaptive step. The classifier output provides a probabilistic measure of flow stability, and the Shannon entropy of these predictions is employed as an uncertainty indicator. KRnet is trained to approximate a probability density function that concentrates sampling in regions of high entropy, thereby directing computational effort towards the evolving bifurcation boundary. This coupling between classification and generative modeling establishes a feedback-driven adaptive learning process analogous to error-indicator based refinement in contemporary partial differential equation solution strategies. Starting from a uniform parameter distribution, the new approach achieves accurate bifurcation boundary identification with significantly fewer Navier--Stokes simulations, providing a scalable foundation for high-dimensional stability analysis.

physics.flu-dyn↗

arXiv2024

An efficient wavelet-based physics-informed neural network for multiscale problems

Physics-informed neural networks (PINNs) are a class of deep learning models that utilize physics in the form of differential equations to address complex problems, including those with limited data availability. However, solving differential equations with rapid oscillations, steep gradients, or singular behavior remains challenging for PINNs. To address this, we propose an efficient wavelet-based physics-informed neural network (W-PINN) that learns solutions in wavelet space. Here, we represent the solution using localized wavelets. This framework represents the solution of a differential equation with significantly fewer degrees of freedom while retaining the dynamics of complex physical phenomena. The proposed architecture enables training to search for solutions within the wavelet domain, where multiscale characteristics are less pronounced compared to the physical domain. This facilitates more efficient training for such problems. Furthermore, the proposed model does not rely on automatic differentiation for derivatives in the loss function and does not require prior information regarding the behavior of the solution, such as the location of abrupt features. The removal of AD significantly reduces training time while maintaining accuracy. Thus, through a strategic fusion of wavelets with PINNs, W-PINNs capture localized nonlinear information, making them well-suited for problems with abrupt behavior, such as singularly perturbed and other multiscale problems. We further analyze the convergence behavior of W-PINN through a comparative study using Neural Tangent Kernel theory. The efficiency and accuracy of the proposed model are demonstrated across various problems, including the FitzHugh--Nagumo (FHN) model, Helmholtz equation, Maxwell equation, Allen--Cahn equation, and lid-driven cavity flow, along with other highly singularly perturbed nonlinear differential equations.

cs.LG↗

arXiv2023

Error analysis of a high-order fully discrete method for two-dimensional time-fractional convection-diffusion equations exhibiting weak initial singularity

This study presents a novel high-order numerical method designed for solving the two-dimensional time-fractional convection-diffusion (TFCD) equation. The Caputo definition is employed to characterize the time-fractional derivative. A weak singularity at the initial time ($t=0$) is encountered in the considered problem, which is effectively managed by adopting a discretization approach for the time-fractional derivative, where Alikhanov's high-order L2-1$_σ$ formula is applied on a non-uniform fitted mesh, resulting in successful tackling of the singularity. A high-order two-dimensional compact operator is implemented to approximate the spatial variables. The alternating direction implicit (ADI) approach is then employed to solve the resulting system of equations by decomposing the two-dimensional problem into two separate one-dimensional problems. The theoretical analysis, encompassing both stability and convergence aspects, has been conducted comprehensively, and it has shown that method is convergent with an order $\mathcal O\left(N_t^{-\min\{3-α,θα,1+2α,2+α\}}+h_x^4+h_y^4\right)$, where $α\in(0,1)$ represents the order of the fractional derivative, $N_t$ is the temporal discretization parameter and $h_x$ and $h_y$ represent spatial mesh widths. Moreover, the parameter $θ$ is utilized in the construction of the fitted mesh.

math.NA↗

arXiv2022

An efficient numerical method based on exponential B-splines for time-fractional Black-Scholes equation governing European options

In this paper a time-fractional Black-Scholes model (TFBSM) is considered to study the price change of the underlying fractal transmission system. We develop and analyze a numerical method to solve the TFBSM governing European options. The numerical method combines the exponential B-spline collocation to discretize in space and a finite difference method to discretize in time. The method is shown to be unconditionally stable using von-Neumann analysis. Also, the method is proved to be convergent of order two in space and $2-μ$ is time, where $μ$ is order of the fractional derivative. We implement the method on various numerical examples in order to illustrate the accuracy of the method, and validation of the theoretical findings. In addition, as an application, the method is used to price several different European options such as the European call option, European put option, and European double barrier knock-out call option.

math.NA↗