arXiv subjects

Siddharth Rout

Publications and source records attributed to Siddharth Rout.

11 recordsLinked to original sources

arXiv2026

Sparse Random-Feature Neural Networks with Krylov-Based SVD for Singularly Perturbed ODE

Random-feature neural networks (RFNNs), including architectures with fixed hidden layers and analytically determined output weights, offer fast training but often suffer from issues due to dense representations of the hidden layer activation. Their reliance on dense feature mappings and least squares solvers can limit scalability and numerical stability, particularly for high-dimensional or stiff systems. Specifically, the activation matrix is observed to be low-rank and extremely ill-conditioned. In this work, we propose a sparse framework for RFNNs that integrates structured sparsity into the hidden layer activations that increases the rank and employs Sparse Singular Value Decomposition (sSVD) for solving the resulting linear least squares problem scalably and efficiently while catering to the bad condition number. We explore the theory behind Lanczos-Golub-Kahan Bidiagonalization technique for sparse SVD and conduct some experiments to identify some limitations and justify the requirement for orthogonalization step in our application. Then, we demonstrate that the proposed method maintains or improves solution accuracy for solving the benchmark one-dimensional steady convection-diffusion equations case having stronger advection, while achieving substantial gains in training efficiency and robustness compared to standard dense implementations.

math.NA↗

arXiv2026

PDE-SSM: A Spectral State Space Approach to Spatial Mixing in Diffusion Transformers

The success of vision transformers-especially for generative modeling-is limited by the quadratic cost and weak spatial inductive bias of self-attention. We propose PDE-SSM, a spatial state-space block that replaces attention with a learnable convection-diffusion-reaction partial differential equation. This operator encodes a strong spatial prior by modeling information flow via physically grounded dynamics rather than all-to-all token interactions. Solving the PDE in the Fourier domain yields global coupling with near-linear complexity of $O(N \log N)$, delivering a principled and scalable alternative to attention. We integrate PDE-SSM into a flow-matching generative model to obtain the PDE-based Diffusion Transformer PDE-SSM-DiT. Empirically, PDE-SSM-DiT matches or exceeds the performance of state-of-the-art Diffusion Transformers while substantially reducing compute. Our results show that, analogous to 1D settings where SSMs supplant attention, multi-dimensional PDE operators provide an efficient, inductive-bias-rich foundation for next-generation vision models.

cs.LG↗

arXiv2026

Learning to Advect: A Neural Semi-Lagrangian Architecture for Weather Forecasting

Recent machine-learning approaches to weather forecasting often employ a monolithic architecture in which distinct physical mechanisms-advection (long-range transport), diffusion-like mixing, thermodynamic processes, and forcing-are represented implicitly within a single large network. This is particularly problematic for advection, where long-range transport typically requires expensive global interaction mechanisms or deep stacks of local convolutional layers. To mitigate this, we present PARADIS, a physics-inspired global weather prediction model that enforces inductive biases on network behavior through a functional decomposition into advection, diffusion, and reaction blocks acting on latent variables. We implement advection through a Neural Semi-Lagrangian operator that performs trajectory-based transport via differentiable interpolation on the sphere, enabling end-to-end learning of both the latent modes to be transported and their characteristic trajectories. Diffusion-like processes are modeled by depthwise-separable spatial mixing, whereas local source terms and vertical interactions are handled via pointwise channel interactions, yielding a physically structured operator decomposition. Evaluated on ERA5 benchmarks, PARADIS achieves competitive deterministic forecast skill, with particularly strong short-lead performance, while preserving substantially better spectral fidelity and forecast activity during medium-range rollouts.

cs.LG↗

arXiv2025

Fast, Convex and Conditioned Network for Multi-Fidelity Vectors and Stiff Univariate Differential Equations

Accuracy in neural PDE solvers often breaks down not because of limited expressivity, but due to poor optimisation caused by ill-conditioning, especially in multi-fidelity and stiff problems. We study this issue in Physics-Informed Extreme Learning Machines (PIELMs), a convex variant of neural PDE solvers, and show that asymptotic components in governing equations can produce highly ill-conditioned activation matrices, severely limiting convergence. We introduce Shifted Gaussian Encoding, a simple yet effective activation filtering step that increases matrix rank and expressivity while preserving convexity. Our method extends the solvable range of Peclet numbers in steady advection-diffusion equations by over two orders of magnitude, achieves up to six orders lower error on multi-frequency function learning, and fits high-fidelity image vectors more accurately and faster than deep networks with over a million parameters. This work highlights that conditioning, not depth, is often the bottleneck in scientific neural solvers and that simple architectural changes can unlock substantial gains.

cs.LG↗

arXiv2025

Fast and Flexible Probabilistic Forecasting of Dynamical Systems using Flow Matching and Physical Perturbation

Learning dynamical systems from incomplete or noisy data is inherently ill-posed, as a single observation may correspond to multiple plausible futures. While physics-based ensemble forecasting relies on perturbing initial states to capture uncertainty, standard Gaussian or uniform perturbations often yield unphysical initial states in high-dimensional systems. Existing machine learning approaches address this via diffusion models, which rely on inference via computationally expensive stochastic differential equations (SDEs). We introduce a novel framework that decouples perturbation generation from propagation. First, we propose a flow matching-based generative approach to learn physically consistent perturbations of the initial conditions, avoiding artifacts caused by Gaussian noise. Second, we employ deterministic flow matching models with Ordinary Differential Equation (ODE) integrators for efficient ensemble propagation with fewer integration steps. We validate our method on nonlinear dynamical system benchmarks, including the Lotka-Volterra Predator-Prey system, MovingMNIST, and high-dimensional WeatherBench data (5.625$^\circ$). Our approach achieves state-of-the-art probabilistic scoring, as measured by the Continuous Ranked Probability Score (CRPS), and physical consistency, while offering significantly faster inference than diffusion-based baselines.

cs.LG↗

arXiv2025

Probabilistic Forecasting for Dynamical Systems with Missing or Imperfect Data

The modeling of dynamical systems is essential in many fields, but applying machine learning techniques is often challenging due to incomplete or noisy data. This study introduces a variant of stochastic interpolation (SI) for probabilistic forecasting, estimating future states as distributions rather than single-point predictions. We explore its mathematical foundations and demonstrate its effectiveness on various dynamical systems, including the challenging WeatherBench dataset.

physics.comp-ph↗

arXiv2024

Advection Augmented Convolutional Neural Networks

Many problems in physical sciences are characterized by the prediction of space-time sequences. Such problems range from weather prediction to the analysis of disease propagation and video prediction. Modern techniques for the solution of these problems typically combine Convolution Neural Networks (CNN) architecture with a time prediction mechanism. However, oftentimes, such approaches underperform in the long-range propagation of information and lack explainability. In this work, we introduce a physically inspired architecture for the solution of such problems. Namely, we propose to augment CNNs with advection by designing a novel semi-Lagrangian push operator. We show that the proposed operator allows for the non-local transformation of information compared with standard convolutional kernels. We then complement it with Reaction and Diffusion neural components to form a network that mimics the Reaction-Advection-Diffusion equation, in high dimensions. We demonstrate the effectiveness of our network on a number of spatio-temporal datasets that show their merit.

cs.LG↗

arXiv2023

Acoustics-based Active Control of Unsteady Flow Dynamics using Reinforcement Learning Driven Synthetic Jets

Flow generated noise are caused shear flows and, hence, they can be used as feedback to control the flow. Existing flow control uses state variables like velocity, pressure, or vorticity, none use acoustic observables as the primary control signal. It is tough to model a classical control algorithm using sound level but data-driven approaches are not as do not have to explicitly model the physics. We present an acoustics-driven framework for active control of unsteady wake dynamics behind a circular cylinder, in which sound is used as the primary feedback signal for flow regulation. The approach integrates deep reinforcement learning (DRL) with synthetic jet actuation, using acoustic measurements acquired from a downstream hydrophone array to inform control decisions in real time. Unlike conventional flow control strategies that rely on velocity or pressure field sensing, the proposed method establishes a direct link between far-field acoustic emissions and near-field actuation. Within this formulation, the DRL agent learns control policies that exploit acoustic signatures of vortex shedding to modulate synthetic jet actuation on the cylinder surface. The resulting control suppresses coherent wake structures and mitigates flow-induced disturbances. Quantitative results show reductions of up to 9.5\% in radiated noise and 23.8\% in drag under the tested conditions, accompanied by a marked attenuation of wake oscillations, for a DFG 2D benchmark flow with Reynolds number 100. These findings demonstrate that acoustic sensing alone can provide sufficient information for effective closed-loop flow control and highlight its potential as a non-intrusive feedback modality for coupled aerodynamic and aeroacoustic optimization in bluff-body flows. The codes for the algorithm can be found here: https://github.com/Siddharth-Rout/FlowControlDRL.

physics.flu-dyn↗

arXiv2023

PINN for Dynamical Partial Differential Equations is Not Training Deeper Networks Rather Learning Advection and Time Variance

The concepts and techniques of physics-informed neural networks (PINNs) is studied and limitations are identified to make it efficient to approximate dynamical equations. Potential working research domains are explored for increasing the robustness of this technique for the solvability of partial differential equations. It is identified that PINNs potentially fails to stronger advection and longer time duration. Also, optimization function and constraint posing needs to be smarter. Even a shallow network is good for a lot of problems while powerful deeper network fails. Reservoir computing based recurrent neural network architecture is recommended to solve dynamical problems.

physics.comp-ph↗

arXiv2022

Airfoil Shape Optimization using Deep Q-Network

The feasibility of using reinforcement learning for airfoil shape optimization is explored. Deep Q-Network (DQN) is used over Markov's decision process to find the optimal shape by learning the best changes to the initial shape for achieving the required goal. The airfoil profile is generated using Bezier control points to reduce the number of control variables. The changes in the position of control points are restricted to the direction normal to the chordline so as to reduce the complexity of optimization. The process is designed as a search for an episode of change done to each control point of a profile. The DQN essentially learns the episode of best changes by updating the temporal difference of the Bellman Optimality Equation. The drag and lift coefficients are calculated from the distribution of pressure coefficient along the profile computed using XFoil potential flow solver. These coefficients are used to give a reward to every change during the learning process where the ultimate aim stands to maximize the cumulate reward of an episode.

cs.LG↗

arXiv2021

Numerical Approximation in CFD Problems Using Physics Informed Machine Learning

The thesis focuses on various techniques to find an alternate approximation method that could be universally used for a wide range of CFD problems but with low computational cost and low runtime. Various techniques have been explored within the field of machine learning to gauge the utility in fulfilling the core ambition. Steady advection diffusion problem has been used as the test case to understand the level of complexity up to which a method can provide solution. Ultimately, the focus stays over physics informed machine learning techniques where solving differential equations is possible without any training with computed data. The prevalent methods by I.E. Lagaris et.al. and M. Raissi et.al are explored thoroughly. The prevalent methods cannot solve advection dominant problems. A physics informed method, called as Distributed Physics Informed Neural Network (DPINN), is proposed to solve advection dominant problems. It increases the lexibility and capability of older methods by splitting the domain and introducing other physics-based constraints as mean squared loss terms. Various experiments are done to explore the end to end possibilities with the method. Parametric study is also done to understand the behavior of the method to different tunable parameters. The method is tested over steady advection-diffusion problems and unsteady square pulse problems. Very accurate results are recorded. Extreme learning machine (ELM) is a very fast neural network algorithm at the cost of tunable parameters. The ELM based variant of the proposed model is tested over the advection-diffusion problem. ELM makes the complex optimization simpler and Since the method is non-iterative, the solution is recorded in a single shot. The ELM based variant seems to work better than the simple DPINN method. Simultaneously scope for various development in future are hinted throughout the thesis.

cs.LG↗