arXiv subjects

Abhisek Ganguly

Publications and source records attributed to Abhisek Ganguly.

5 recordsLinked to original sources

arXiv2026

Deep Neural Networks as Discrete Dynamical Systems: Implications for Physics-Informed Learning

We revisit the analogy between feed-forward deep neural networks (DNNs) and discrete dynamical systems derived from neural integral equations and their corresponding partial differential equation (PDE) forms. A comparative analysis between the numerical/exact solutions of the Burgers' and Eikonal equations, and the same obtained via PINNs is presented. We show that PINN learning provides a different computational pathway compared to standard numerical discretization in approximating essentially the same underlying dynamics of the system. Within this framework, DNNs can be interpreted as discrete dynamical systems whose layer-wise evolution approaches attractors, and multiple parameter configurations may yield comparable solutions, reflecting the degeneracy of the inverse mapping. In contrast to the structured operators associated with finite-difference (FD) procedures, PINNs learn dense parameter representations that are not directly associated with classical discretization stencils. This distributed representation generally involves a larger number of parameters, leading to reduced interpretability and increased computational cost. However, the additional flexibility of such representations may offer advantages in high-dimensional settings where classical grid-based methods become impractical.

cs.LG↗

arXiv2026

Randomness and signal propagation in physics-informed neural networks (PINNs): A neural PDE perspective

Physics-informed neural networks (PINNs) often exhibit weight matrices that appear statistically random after training, yet their implications for signal propagation and stability remain unsatisfactorily understood, let alone the interpretability. In this work, we analyze the spectral and statistical properties of trained PINN weights using viscous and inviscid variants of the one-dimensional Burgers' equation, and show that the learned weights reside in a high-entropy regime consistent with predictions from random matrix theory. To investigate the dynamical consequences of such weight structures, we study the evolution of signal features inside a network through the lens of neural partial differential equations (neural PDEs). We show that random and structured weight matrices can be associated with specific discretizations of neural PDEs, and that the numerical stability of these discretizations governs the stability of signal propagation through the network. In particular, explicit unstable schemes lead to degraded signal evolution, whereas stable implicit and higher-order schemes yield well-behaved dynamics for the same underlying neural PDE. Our results offer an explicit example of how numerical stability and network architecture shape signal propagation in deep networks, in relation to random matrix and neural PDE descriptions in PINNs.

cs.LG↗

arXiv2026

Kinetic-based regularization: Learning spatial derivatives and PDE applications

Accurate estimation of spatial derivatives from discrete and noisy data is central to scientific machine learning and numerical solutions of PDEs. We extend kinetic-based regularization (KBR), a localized multidimensional kernel regression method with a single trainable parameter, to learn spatial derivatives with provable second-order accuracy in 1D. Two derivative-learning schemes are proposed: an explicit scheme based on the closed-form prediction expressions, and an implicit scheme that solves a perturbed linear system at the points of interest. The fully localized formulation enables efficient, noise-adaptive derivative estimation without requiring global system solving or heuristic smoothing. Both approaches exhibit quadratic convergence, matching second-order finite difference for clean data, along with a possible high-dimensional formulation. Preliminary results show that coupling KBR with conservative solvers enables stable shock capture in 1D hyperbolic PDEs, acting as a step towards solving PDEs on irregular point clouds in higher dimensions while preserving conservation laws.

math.NA↗

arXiv2025

Dual Computational Horizons: Incompleteness and Unpredictability in Intelligent Systems

We formalize two independent computational limitations that constrain algorithmic intelligence: formal incompleteness and dynamical unpredictability. The former limits the deductive power of consistent reasoning systems while the latter bounds long-term prediction under finite precision. We show that these two extrema together impose structural bounds on an agent's ability to reason about its own predictive capabilities. In particular, an algorithmic agent cannot verify its own maximal prediction horizon universally. This perspective clarifies inherent trade-offs between reasoning, prediction, and self-analysis in intelligent systems. The construction presented here constitutes one representative instance of a broader logical class of such limitations.

cs.AI↗

arXiv2025

A kinetic-based regularization method for data science applications

We propose a physics-based regularization technique for function learning, inspired by statistical mechanics. By drawing an analogy between optimizing the parameters of an interpolator and minimizing the energy of a system, we introduce corrections that impose constraints on the lower-order moments of the data distribution. This minimizes the discrepancy between the discrete and continuum representations of the data, in turn allowing to access more favorable energy landscapes, thus improving the accuracy of the interpolator. Our approach improves performance in both interpolation and regression tasks, even in high-dimensional spaces. Unlike traditional methods, it does not require empirical parameter tuning, making it particularly effective for handling noisy data. We also show that thanks to its local nature, the method offers computational and memory efficiency advantages over Radial Basis Function interpolators, especially for large datasets.

cs.LG↗