arXiv subjects

B. Veena S. N. Rao

Publications and source records attributed to B. Veena S. N. Rao.

7 recordsLinked to original sources

arXiv2026

Mesh-Free Numerical Approximation of the Biharmonic Equation via Optimized Kolmogorov--Arnold Neural Networks

A mesh-free numerical framework based on \textit{Kolmogorov--Arnold Physics-Informed Neural Networks} (KAN-PINNs) is developed for the approximation of fourth-order elliptic boundary value problems, with specific application to the biharmonic equation governing thin plate deflection. Unlike conventional Multi-Layer Perceptrons relying on fixed nodal activations, learnable univariate functions parameterized via radial basis functions are deployed on network edges, while high-order differentiability is preserved through hyperbolic tangent activation mappings. The severe numerical stiffness inherent to fourth-order differential operators and fully clamped boundary conditions is addressed through a direct normalized residual formulation coupled with a hybrid, multi-stage AdamW-to-L-BFGS optimization pipeline. An automated \textit{24/7 hill-climbing search protocol} is implemented to systematically calibrate boundary penalty weights and optimization schedules. When evaluated on a smooth manufactured benchmark on the unit square, an error reduction factor exceeding $150\times$ is attained over successive iterations, culminating in a final relative $L_2$ error of $1.593 \times 10^{-5}$ ($0.00159\%$) and a training loss of $3.065 \times 10^{-6}$ utilizing only $7,801$ trainable parameters. These results demonstrate that high-order PDE problems can be accurately resolved using compact, mesh-free KAN-PINNs architecture without requiring auxiliary variable transformations or discrete mesh generation.

math.NA↗

arXiv2026

Neural Network Solvers for Nonlinear ODEs: Benchmarking MLPs and B-Spline/Chebyshev KANs

Three neural networks are compared as solvers of nonlinear ordinary differential equations: a multilayer perceptron (MLP), a Kolmogorov-Arnold network (KAN) whose edges are B-splines, and a KAN whose edges are Chebyshev series. Each network is trained only on the residual of the equation and on its initial or boundary conditions. The exact solution, or a numerical solution accurate to $10^{-13}$, is used only afterward to measure the error. Five test problems of differential orders one to four are investigated: the logistic equation, a singular Lane-Emden type initial value problem, a singular boundary value problem, the Falkner-Skan equation with suction, a fourth-order problem. The same training budget and the same choice of five activation functions are allocated to all networks. No numerical failures are encountered in any of the \nRuns{} training runs. For every problem, the best KAN is found to be at least as accurate as the best MLP while \paramsKANmin{} to \paramsKANmax{} parameters are used instead of \paramsMLP{}; the margin ranges from a tie to a factor of three. In the Falkner-Skan setting of an earlier study, the B-spline KAN is shown to be 34 to 220 times more accurate at every published test point. Much smaller MLPs are also found to be competitive, whereas at orders three and four, failures are experienced more frequently by KANs, and a coarse basis is required. The derivatives in the equation are distorted by the usual grid extension of a B-spline KAN; consequently, it is replaced with knot insertion, by which the logistic error is lowered more than a hundredfold, though no improvement is provided at order four.

math.GM↗

arXiv2026

AKAPINN: Adaptive Kolmogorov-Arnold Physics-Informed Neural Networks for approximating solutions to quasilinear partial differential equations

A controlled comparative study of a mesh-free Kolmogorov--Arnold Physics-Informed Neural Network (KAN-PINN) applied to nonlinear strain-limiting partial differential equations is presented. Three distinct training runs across varying material-parameter pairs $(α, β)$ are evaluated within a fixed computational pipeline comprising an exact hard boundary ansatz, a robust pseudo-Huber residual loss, and a multi-stage Adam-to-L-BFGS optimizer schedule. A fundamental metric inversion is revealed by the analysis: while training loss reflects operator-specific residual scaling and constitutive stiffness, the operator-independent relative $L_2$ error demonstrates that intermediate parameter regimes achieve superior solution accuracy and enhanced generalization.

math.NA↗

arXiv2025

A Mechanistic Framework for in Silico Optimization of Neuroblastoma Chemo-Immunotherapy

A critical need exists for optimal therapeutic strategies for neuroblastoma, a prevalent and often fatal pediatric solid malignancy. To address the demand for quantitative models that can guide clinical decision-making, a novel mathematical framework was developed. Combination therapies involving immunotherapy, such as Interleukin-2 (IL-2), and chemotherapy, exemplified by Cyclophosphamide, have shown significant clinical potential by enhancing anti-tumor immune responses. In this study, a nonlinear system of coupled ordinary differential equations was formulated to mechanistically describe the interactions among tumor cells, natural killer (NK) cells, and cytotoxic T lymphocytes (CTLs). The pharmacodynamic effects of both IL-2 and Cyclophosphamide on these key immune populations were explicitly incorporated, allowing for the simulation of tumor dynamics across distinct patient risk profiles. The resulting computational framework provides a robust platform for the \textit{\textbf{in silico}} \textbf{optimization} of therapeutic regimens, presenting a quantitative pathway toward the improvement of clinical outcomes for patients with neuroblastoma.

q-bio.TO↗

arXiv2025

Numerical Approximation of Electrohydrodynamics Model: A Comparative Study of PINNs and FEM

The accurate representation of numerous physical, chemical, and biological processes relies heavily on differential equations (DEs), particularly nonlinear differential equations (NDEs). While understanding these complex systems necessitates obtaining solutions to their governing equations, the derivation of precise approximations for NDEs remains a formidable task in computational mathematics. Although established techniques such as the finite element method (FEM) have long been foundational, remarkable promise for approximating continuous functions with high efficacy has recently been demonstrated by advancements in physics-informed deep-learning feedforward neural networks. In this work, a novel application of PINNs is presented for the approximation of the challenging Electrohydrodynamic (EHD) problem. A specific $L^2$-type \textit{total loss function} is employed, notably without reliance on any prior knowledge of the exact solution. A comprehensive comparative study is conducted, juxtaposing the approximation capabilities of the proposed neural network with those of the conventional FEM. The PINN training regimen is composed of two critical steps: forward propagation for adjustments to gradient and curvature, and backpropagation for the refinement of hyperparameters. The critical challenge of identifying optimal neural network architectures and hyperparameter configurations for efficient optimization is meticulously investigated. Excellent performance is shown to be delivered by the neural network even with a limited training dataset. Simultaneously, it is demonstrated that the accuracy of the FEM can be substantially enhanced through the judicious selection of smaller mesh sizes.

math.NA↗

arXiv2025

Neural solver for sixth-order ordinary differential equations

A method for approximating sixth-order ordinary differential equations is proposed, which utilizes a deep learning feedforward artificial neural network, referred to as a neural solver. The efficacy of this unsupervised machine learning method is demonstrated through the solution of two distinct boundary value problems (BVPs), with the method being extended to include the solution of a sixth-order ordinary differential equation (ODE). The proposed mean squared loss function is comprised of two terms: the differential equation is satisfied by the first term, while the initial or boundary conditions are satisfied by the second. The total loss function is minimized using a quasi-Newton optimization method to obtain the desired network output. The approximation capability of the proposed method is verified for sixth-order ODEs. Point-wise comparisons of the approximations show strong agreement with available exact solutions. The proposed algorithm minimizes the overall learnable network hyperparameters in a given BVP. Simple minimization of the total loss function yields highly accurate results even with a low number of epochs. Therefore, the proposed framework offers an attractive setting for the computational mathematics community.

math.NA↗

arXiv2025

Automated $h$-adaptivity for finite element approximations of the Falkner-Skan equation

This paper details the development and application of an $h$-adaptive finite element method for the numerical solution of the \textit{Falkner-Skan equation}. A posteriori error estimation governs the adaptivity of the mesh, specifically the well-established \textit{Kelly error estimator}, which utilizes the jump in the gradient across elements. The implementation of this method allowed for accurate and efficient resolution of the boundary layer behavior characteristic of Falkner-Skan flows. Numerical solutions were obtained across various wedge flow parameters, encompassing favorable and adverse pressure gradients. A key focus of this study was the precise computation of the skin friction coefficient, a critical parameter in boundary layer analysis, across this diverse range of flow conditions. The results are presented and discussed, demonstrating the robustness and accuracy of the adaptive finite element approach for this class of nonlinear boundary layer problems.

math.NA↗