Siddharth Rout

Eshed Gal, Moshe Eliasof, Siddharth Rout et al.

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.

Siddharth Rout, Chao-An Lin

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.

Carlos A. Pereira, Stéphane Gaudreault, Valentin Dallerit et al.

Recent machine-learning approaches to weather forecasting often employ a monolithic architecture, where distinct physical mechanisms (advection, transport), diffusion-like mixing, thermodynamic processes, and forcing are represented implicitly within a single large network. This representation is particularly problematic for advection, where long-range transport must be treated with expensive global interaction mechanisms or through deep, stacked convolutional layers. To mitigate this, we present PARADIS, a physics-inspired global weather prediction model that imposes 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 through depthwise-separable spatial mixing, while local source terms and vertical interactions are modeled via pointwise channel interactions, enabling operator-level physical structure. PARADIS provides state-of-the-art forecast skill at a fraction of the training cost. On ERA5-based benchmarks, the 1 degree PARADIS model, with a total training cost of less than a GPU month, meets or exceeds the performance of 0.25 degree traditional and machine-learning baselines, including the ECMWF HRES forecast and DeepMind's GraphCast.

Kevin Kurian Thomas Vaidyan, Siddharth Rout

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.

Siddharth Rout

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.

Siddharth Rout, Eldad Haber, Stéphane Gaudreault

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.

Siddharth Rout, Eldad Haber, Stephane Gaudreault

Learning dynamical systems is crucial across many fields, yet applying machine learning techniques remains challenging due to missing variables and noisy data. Classical mathematical models often struggle in these scenarios due to the arose ill-posedness of the physical systems. Stochastic machine learning techniques address this challenge by enabling the modeling of such ill-posed problems. Thus, a single known input to the trained machine learning model may yield multiple plausible outputs, and all of the outputs are correct. In such scenarios, probabilistic forecasting is inherently meaningful. In this study, we introduce a variant of flow matching for probabilistic forecasting which estimates possible future states as a distribution over possible outcomes rather than a single-point prediction. Perturbation of complex dynamical states is not trivial. Community uses typical Gaussian or uniform perturbations to crucial variables to model uncertainty. However, not all variables behave in a Gaussian fashion. So, we also propose a generative machine learning approach to physically and logically perturb the states of complex high-dimensional dynamical systems. Finally, we establish the mathematical foundations of our method and demonstrate its effectiveness on several challenging dynamical systems, including a variant of the high-dimensional WeatherBench dataset, which models the global weather at a 5.625° meridional resolution.

Niloufar Zakariaei, Siddharth Rout, Eldad Haber et al.

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.

Siddharth Rout, Vikas Dwivedi, Balaji Srinivasan

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.