arXiv subjects

Pankaj K Mishra

Publications and source records attributed to Pankaj K Mishra.

12 recordsLinked to original sources

arXiv2026

Backend-agnostic Julia framework for 3D modeling and inversion of gravity data

This paper presents a high-performance framework for three-dimensional gravity modeling and inversion implemented in Julia, addressing key computational challenges associated with gravity inversion, including large scale problem, ill-posedness, and non-uniqueness. The framework employs a data-space inversion formulation that reduces the dimensionality of the inverse problem, resulting in lower memory requirements and improved computational efficiency. Forward modeling and inversion operators are implemented using a backend-agnostic kernel abstraction, allowing the same computational code to run on multicore CPUs and GPU accelerators. Performance evaluations using NVIDIA CUDA GPUs show significant reductions in computational time relative to CPU execution, particularly for large-scale problems involving up to approximately 3.3 million rectangular prisms and 178,797 gravity observations. The inversion incorporates implicit model constraints through the data-space formulation and depth-weighted sensitivity, which reduces the effect of depth-dependent amplitude decay and promotes geologically coherent density models. Synthetic experiments demonstrate the capability of the framework to recover complex subsurface structures, including vertical and dipping dykes. Application to field gravity data further demonstrates the practical applicability of the proposed approach, with the recovered density distributions showing good agreement with independent geological constraints and previous interpretations. The results demonstrate that GPU-accelerated Julia provides an efficient and extensible platform for large-scale three-dimensional gravity modeling and inversion, enabling high-resolution geophysical investigations with reduced computational and memory requirements.

physics.geo-ph↗

arXiv2026

Implicit Neural Representations Framework for One-Dimensional Magnetotelluric Inversion

Magnetotelluric (MT) inversion is a very useful technique to image the subsurface electrical resistivity structures. It is used for mineral exploration, geothermal studies, groundwater assessment, and lithospheric investigations. In this work, we proposed a physics-informed machine learning framework for 1D MT inversion based on implicit neural representations (INR). Our approach models the subsurface resistivity as a continuous function of depth using a coordinate-based neural network. This method does not require fixed discretization or layered models. The neural network is trained directly on a differentiable MT forward-model loss based on Wait's recursive impedance formulation. This setup allows inversion to occur in a physics-consistent optimization framework. The implicit regularization avoids the need for manual tuning of external regularization. We have tested this method on synthetic conductor models and real MT data. The results showed its ability to recover geologically relevant resistivity structures over various depths and thicknesses. Through different initializations, we can compute an ensemble of plausible models to estimate model uncertainty. These results suggest that implicit neural representations provide a flexible framework for geophysical inversion, with even greater potential in higher-dimensional MT problems and joint inversion applications.

physics.geo-ph↗

arXiv2026

Three-dimensional inversion of gravity data using implicit neural representations and scientific machine learning

Inversion of gravity data is an important method for investigating subsurface density variations relevant to mineral exploration, geothermal assessment, carbon storage, natural hydrogen, groundwater resources, and tectonic evolution. Here we present a scientific machine-learning approach for three-dimensional gravity inversion that represents subsurface density as a continuous field using an implicit neural representation (INR). The method trains a deep neural network directly through a physics-based forward-model loss, mapping spatial coordinates to a continuous density field without predefined meshes or discretisation. Spatial encoding enhances the network's capacity to capture sharp contrasts and short-wavelength features that conventional coordinate-based networks tend to oversmooth due to spectral bias. We demonstrate the approach on synthetic examples including smooth models, representing realistic geological complexity, and a dipping block model to assess recovery of structures at different depths. The INR framework reconstructs detailed structure and geologically plausible boundaries without explicit regularisation or depth weighting, while reducing the number of inversion parameters as the problem size grows bigger. These results highlight the potential of implicit representations to enable scalable, flexible, and interpretable large-scale geophysical inversion. This framework could generalise to other geophysical methods and for joint/multiphysics inversion.

physics.geo-ph↗

arXiv2024

Enhancing RBF-FD Efficiency for Highly Non-Uniform Node Distributions via Adaptivity

Radial basis function generated finite-difference (RBF-FD) methods have recently gained popularity due to their flexibility with irregular node distributions. However, the convergence theories in the literature, when applied to nonuniform node distributions, require shrinking fill distance and do not take advantage of areas with high data density. Non-adaptive approach using same stencil size and degree of appended polynomial will have higher local accuracy at high density region, but has no effect on the overall order of convergence and could be a waste of computational power. This work proposes an adaptive RBF-FD method that utilizes the local data density to achieve a desirable order accuracy. By performing polynomial refinement and using adaptive stencil size based on data density, the adaptive RBF-FD method yields differentiation matrices with higher sparsity while achieving the same user-specified convergence order for nonuniform point distributions. This allows the method to better leverage regions with higher node density, maintaining both accuracy and efficiency compared to standard non-adaptive RBF-FD methods.

math.NA↗

arXiv2023

Approximation- and Chattering-free Quasi Sliding Mode Control for Unknown Systems

Motivated by the concept of Quasi-Sliding Mode (QSM) in discrete-time systems, this paper presents a novel approach that relaxes the requirement of ubiquitous exact sliding motion in continuous-time systems, aiming to achieve an approximation- and chattering-free quasi-sliding mode controller (QSMC). The proposed QSMC provides robust and chattering-free control for an unknown nonlinear system without the need for intricate control methods, learning agents, or system identification tools. Furthermore, comprehensive simulation studies have been conducted to validate the effectiveness of the proposed strategy.

eess.SY↗

arXiv2023

Approximation-free control for unknown systems with performance and input constraints

This paper addresses the problem of tracking control for an unknown nonlinear system with time-varying bounded disturbance subjected to prescribed Performance and Input Constraints (PIC). Since simultaneous prescription of PIC involves a trade-off, we propose an analytical feasibility condition to prescribe feasible PIC which also yields feasible initial state space as corollary results. Additionally, an approximation-free controller is proposed to guarantee that the tracking performance adheres to the prescribed PIC. The effectiveness of the proposed approach is demonstrated through numerical examples.

math.OC↗

arXiv2022

Approximation-free Prescribed Performance Control with Prescribed Input Constraints

This paper considers the tracking control problem for an unknown nonlinear system with time-varying bounded disturbance subjected to a prescribed performance and input constraints. When performance and input constraints are specified simultaneously for such a problem, a trade-off is inevitable. Consequently, a feasibility condition for prescribing performance and input constraints is devised to address such difficulties of arbitrary prescription. In addition, an approximation-free controller with low complexity is proposed, which ensures that the constraints are never violated, provided that the feasibility condition holds. Finally, simulation results corroborate the effectiveness of the proposed controller.

eess.SY↗

arXiv2021

RBF-FD analysis of 2D time-domain acoustic wave propagation in heterogeneous media

Radial Basis Function-generated Finite Differences (RBF-FD) is a popular variant of local strong-form meshless methods that do not require a predefined connection between the nodes, making it easier to adapt node-distribution to the problem under consideration. This paper investigates an RBF-FD solution of time-domain acoustic wave propagation in the context of seismic modeling in the Earth's subsurface. Through a number of numerical tests, ranging from homogeneous to highly-heterogeneous velocity models including non-smooth irregular topography, we demonstrate that the present approach can be further generalized to solve large-scale seismic modeling and full waveform inversion problems in arbitrarily complex models enabling more robust interpretations of geophysical observations

cs.CE↗

arXiv2018

A stabilized radial basis-finite difference (RBF-FD) method with hybrid kernels

Recent developments have made it possible to overcome grid-based limitations of finite difference (FD) methods by adopting the kernel-based meshless framework using radial basis functions (RBFs). Such an approach provides a meshless implementation and is referred to as the radial basis-generated finite difference (RBF-FD) method. In this paper, we propose a stabilized RBF-FD approach with a hybrid kernel, generated through a hybridization of the Gaussian and cubic RBF. This hybrid kernel was found to improve the condition of the system matrix, consequently, the linear system can be solved with direct solvers which leads to a significant reduction in the computational cost as compared to standard RBF-FD methods coupled with present stable algorithms. Unlike other RBF-FD approaches, the eigenvalue spectra of differentiation matrices were found to be stable irrespective of irregularity, and the size of the stencils. As an application, we solve the frequency-domain acoustic wave equation in a 2D half-space. In order to suppress spurious reflections from truncated computational boundaries, absorbing boundary conditions have been effectively implemented.

math.NA↗

arXiv2018

Hybrid Gaussian-cubic radial basis functions for scattered data interpolation

Scattered data interpolation schemes using kriging and radial basis functions (RBFs) have the advantage of being meshless and dimensional independent, however, for the data sets having insufficient observations, RBFs have the advantage over geostatistical methods as the latter requires variogram study and statistical expertise. Moreover, RBFs can be used for scattered data interpolation with very good convergence, which makes them desirable for shape function interpolation in meshless methods for numerical solution of partial differential equations. For interpolation of large data sets, however, RBFs in their usual form, lead to solving an ill-conditioned system of equations, for which, a small error in the data can cause a significantly large error in the interpolated solution. In order to reduce this limitation, we propose a hybrid kernel by using the conventional Gaussian and a shape parameter independent cubic kernel. Global particle swarm optimization method has been used to analyze the optimal values of the shape parameter as well as the weight coefficients controlling the Gaussian and the cubic part in the hybridization. Through a series of numerical tests, we demonstrate that such hybridization stabilizes the interpolation scheme by yielding a far superior implementation compared to those obtained by using only the Gaussian or cubic kernels. The proposed kernel maintains the accuracy and stability at small shape parameter as well as relatively large degrees of freedom, which exhibit its potential for scattered data interpolation and intrigues its application in global as well as local meshless methods for numerical solution of PDEs.

math.NA↗

arXiv2017

An improved radial basis-pseudospectral method with hybrid Gaussian-cubic kernels

While pseudospectral (PS) methods can feature very high accuracy, they tend to be severely limited in terms of geometric flexibility. Application of global radial basis functions overcomes this, however at the expense of problematic conditioning (1) in their most accurate flat basis function regime, and (2) when problem sizes are scaled up to become of practical interest. The present study considers a strategy to improve on these two issues by means of using hybrid radial basis functions that combine cubic splines with Gaussian kernels. The parameters, controlling Gaussian and cubic kernels in the hybrid RBF, are selected using global particle swarm optimization. The proposed approach has been tested with radial basis-pseudospectral method for numerical approximation of Poisson, Helmholtz, and Transport equation. It was observed that the proposed approach significantly reduces the ill-conditioning problem in the RBF-PS method, at the same time, it preserves the stability and accuracy for very small shape parameters. The eigenvalue spectra of the coefficient matrices in the improved algorithm were found to be stable even at large degrees of freedom, which mimic those obtained in pseudospectral approach. Also, numerical experiments suggest that the hybrid kernel performs significantly better than both pure Gaussian and pure cubic kernels.

math.NA↗

arXiv2015

Meshless RBF based pseudospectral solution of acoustic wave equation

Chebyshev pseudospectral (PS) methods are reported to provide highly accurate solution using polynomial approximation. Use of polynomial basis functions in PS algorithms limits the formulation to univariate systems constraining it to tensor product grids for multi-dimensions. Recent studies have shown that replacing the polynomial by radial basis functions in pseudospectral method (RBF-PS) has the advantage of using irregular grids for multivariate systems. A RBF-PS algorithm has been presented here for the numerical solution of inhomogeneous Helmholtz's equation using Gaussian RBF for derivative approximation. Efficacy of RBF approximated derivatives has been checked through error analysis comparison with PS method. Comparative study of PS, RBF-PS and finite difference approach for the solution of a linear boundary value problem has been performed. Finally, a typical frequency domain acoustic wave propagation problem has been solved using Dirichlet boundary condition and a point source. The algorithm presented here can be extended further for seismic modeling with complexities associated with absorbing boundary conditions.

physics.comp-ph↗