arXiv subjects

Tuhin Sahai

Publications and source records attributed to Tuhin Sahai.

At least 19 recordsLinked to original sources

arXiv2026

Optimized Piecewise Affine Abstractions of Neural Networks with Learnable Activation Functions

We present a generalized framework for the range verification of neural networks featuring non-linear activation functions. Our approach first constructs an ``optimized piecewise affine abstraction" of the network that replaces each non-linear activation function by a piecewise affine (PWA) function plus a bounded error. Such PWA functions are readily amenable to existing neural network verification techniques using specializations of linear arithmetic SMT solvers and mixed-integer optimization approaches. However, there are infinitely many ways to abstract each node, with a natural tradeoff between the number of pieces used, the global error bound, and the complexity of the resulting verification problem. We propose a dynamic programming (DP) algorithm to systematically compute the optimized PWA abstraction for general activation functions, guaranteeing tighter output bounds. The algorithm combines a local DP approximation at each node with a global error bound, yielding a variant of the knapsack problem for deciding how to allocate a fixed budget on the total number of pieces across units so as to minimize the worst-case error bound between the network and its approximation. Although the knapsack problem is itself NP-hard, we can use pseudo-polynomial DP algorithms as well as approximation schemes to solve it efficiently. Crucially, our approach is broadly applicable to diverse networks consisting of non-linear activations, including standard Multi-Layer Perceptrons (MLPs) and recently proposed architectures such as Kolmogorov-Arnold Networks (KANs). Over a series of KAN benchmarks spanning 20 to 22,000 parameters, our approach yields output bounds that are consistently of smaller width than uniform PWA allocation. The overall time taken is roughly comparable while the overhead for computing the optimized abstraction is subsumed by the time taken to compute output bounds.

cs.LG↗

arXiv2026

Sharp Stability Threshold and Certification for Designing Stable Residual Architectures

We propose \emph{the sublinear-growth principle} for deep residual architectures -- a sharp stability threshold on the input-magnitude exponent of every residual block's velocity field: $$\|v(x, t)\| \leq c\,\|x\|^q + b, \qquad q \in [0, 1].$$ The threshold $q = 1$ is established via two independent arguments. Classical ODE theory gives a global forward flow on $[0, T]$ at $q \le 1$ and exhibits divergent velocity fields at any $q > 1$. The optimal-control analysis, via the Hamilton-Jacobi-Bellman equation, sharpens this to a selection statement: the training optimum is bang-bang on the boundary of the admissible class, so the optimum at $q > 1$ blows up while the optimum at $q \le 1$ is safe by construction. The exponent criterion $q \le 1$ is thereby a necessary and sufficient condition for stable training. It clarifies architectural placements that ensure the stability of training and inference, explaining, for instance, the stabilizing role of layer normalization. The sublinear-growth velocity fields form \emph{the right function space} on which forward dynamics, adjoint sensitivity, and architectural composition are all well-controlled. An arithmetic of input-magnitude exponents under the five operations that build residual blocks enables efficient certification of $q_k \le 1$ at the level of architectural primitives, in place of ad hoc trial and error in the search for stable neural architectural designs. A parameter-free modification reduces the supercritical Mamba block from $q = 5$ to $q = 1$ without layer normalization, demonstrating this point. Experiments on Mamba and PatchTST confirm that the $q \le 1$ variants train stably: the criterion is the input-magnitude exponent, not the presence of a normalization layer.

cs.LG↗

arXiv2026

ISOMORPH: A Supply Chain Digital Twin for Simulation, Dataset Generation, and Forecasting Benchmarks

Open time-series forecasting (TSF) benchmarks cover retail, energy, weather, and traffic, but supply-chain logistics remains underserved. We introduce ISOMORPH, the first public digital twin of a multi-echelon logistics network with interpretable, user-configurable parameters and modular topology, demand, and control rules. The simulator advances a directed routing graph in discrete time: demand is served from inventory or recorded as backlog and triggers replenishment throughout the network. The state tracks inventory, outstanding orders, in-transit shipments, and a smoothed demand estimate, yielding Markovian dynamics on a tractable state space. The released data reproduces the bullwhip effect at empirically consistent magnitudes, while three conservation laws provide verification tools for simulator extensions. We release datasets at two catalogue scales ($C=50$ and $C=200$), six scenario sweeps, and 20 Latin-hypercube perturbations. These datasets exhibit dynamics largely absent from fixed TSF benchmarks, including variance amplification, cascading bottlenecks, regime shifts, and cross-channel coupling through shared macro shocks. Zero-shot evaluation of four foundation models (Chronos, Moirai, TimesFM, and Lag-Llama) yields MASE values exceeding public GIFT-Eval references at low-to-moderate horizons, supporting incorporation into existing benchmark suites. The same models provide forecast confidence bands through Latin-hypercube perturbations of demand-side parameters, enabling forward uncertainty quantification (UQ) unavailable on standard TSF datasets and demonstrating that foundation models can serve as fast surrogates for digital-twin-based UQ. Code (MIT): https://github.com/tuhinsahai/ISOMORPH. Interactive demo: https://huggingface.co/spaces/HyeminGu/ISOMORPH-demo.

stat.ML↗

arXiv2026

Vocabulary-size-independent Convergence of Discrete Diffusion Models: adjoint equations induce the right space

Discrete diffusion has become a leading framework for generative modeling in various applications including language, vision, and biology. Existing convergence theory, however, exhibits fundamental limitations. KL-based analyses diverge under singular priors such as the masked distribution, while bounds in total variation (TV) depend on the vocabulary size $S$ and become vacuous for modern language tasks, where vocabularies contain hundreds of thousands of tokens. We develop a unified adjoint-equation-based framework that establishes vocabulary-size-independent convergence guarantees in any integral probability metric (IPM). To the best of our knowledge, our bounds are the first to be entirely free of $S$ and applicable to both masked and uniform priors. Importantly, our results can extend existing step complexity guarantees to any IPM. Also, our theory relies only on a single standard rate-matrix regularity assumption and applies to general priors. Five novel techniques drive our improvements: 1. working in the space of observables via adjoint equations rather than directly with probability measures; 2. a regularity analysis that yields bounds on any IPM; 3. a coupling argument that removes $S$-dependence under uniform transitions; and 4. score-marginal cancellation and 5. exit-routing techniques that remove $S$-dependence under masked transitions. Our framework thus sharply departs from prior analyses and avoids the shortcomings of pathspace-KL and existing TV-based approaches. Beyond convergence bounds, our framework provides a versatile toolkit for further theoretical study of discrete diffusion models, including principled choices of loss functions and vocabulary-size-independent step complexity.

cs.LG↗

arXiv2025

Optimal Control for Transformer Architectures: Enhancing Generalization, Robustness and Efficiency

We study Transformers through the perspective of optimal control theory, using tools from continuous-time formulations to derive actionable insights into training and architecture design. This framework improves the performance of existing Transformer models while providing desirable theoretical guarantees, including generalization and robustness. Our framework is designed to be plug-and-play, enabling seamless integration with established Transformer models and requiring only slight changes to the implementation. We conduct seven extensive experiments on tasks motivated by text generation, sentiment analysis, image classification, and point cloud classification. Experimental results show that the framework improves the test performance of the baselines, while being more parameter-efficient. On character-level text generation with nanoGPT, our framework achieves a 46% reduction in final test loss while using 42% fewer parameters. On GPT-2, our framework achieves a 9.3% reduction in final test loss, demonstrating scalability to larger models. To the best of our knowledge, this is the first work that applies optimal control theory to both the training and architecture of Transformers. It offers a new foundation for systematic, theory-driven improvements and moves beyond costly trial-and-error approaches.

cs.LG↗

arXiv2025

Stability of Transformers under Layer Normalization

Despite their widespread use, training deep Transformers can be unstable. Layer normalization, a standard component, improves training stability, but its placement has often been ad-hoc. In this paper, we conduct a principled study on the forward (hidden states) and backward (gradient) stability of Transformers under different layer normalization placements. Our theory provides key insights into the training dynamics: whether training drives Transformers toward regular solutions or pathological behaviors. For forward stability, we derive explicit bounds on the growth of hidden states in trained Transformers. For backward stability, we analyze how layer normalization affects the backpropagation of gradients, thereby explaining the training dynamics of each layer normalization placement. Our analysis also guides the scaling of residual steps in Transformer blocks, where appropriate choices can further improve stability and performance. Our numerical results corroborate our theoretical findings. Beyond these results, our framework provides a principled way to sanity-check the stability of Transformers under new architectural modifications, offering guidance for future designs.

cs.LG↗

arXiv2025

Discovering Algorithms with Computational Language Processing

Algorithms are the engine for reproducible problem-solving. We present a framework automating algorithm discovery by conceptualizing them as sequences of operations, represented as tokens. These computational tokens are chained using a grammar, enabling the formation of increasingly sophisticated procedures. Our ensemble Monte Carlo tree search (MCTS) guided by reinforcement learning (RL) explores token chaining and drives the creation of new tokens. This methodology rediscovers, improves, and generates new algorithms that substantially outperform existing methods for strongly NP-hard combinatorial optimization problems and foundational quantum computing approaches such as Grover's and Quantum Approximate Optimization Algorithm. Operating at the computational rather than code-generation level, our framework produces algorithms that can be tailored specifically to problem instances, not merely classes.

cs.AI↗

arXiv2025

Taylor-Model Physics-Informed Neural Networks (PINNs) for Ordinary Differential Equations

We study the problem of learning neural network models for Ordinary Differential Equations (ODEs) with parametric uncertainties. Such neural network models capture the solution to the ODE over a given set of parameters, initial conditions, and range of times. Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for learning such models that combine data-driven deep learning with symbolic physics models in a principled manner. However, the accuracy of PINNs degrade when they are used to solve an entire family of initial value problems characterized by varying parameters and initial conditions. In this paper, we combine symbolic differentiation and Taylor series methods to propose a class of higher-order models for capturing the solutions to ODEs. These models combine neural networks and symbolic terms: they use higher order Lie derivatives and a Taylor series expansion obtained symbolically, with the remainder term modeled as a neural network. The key insight is that the remainder term can itself be modeled as a solution to a first-order ODE. We show how the use of these higher order PINNs can improve accuracy using interesting, but challenging ODE benchmarks. We also show that the resulting model can be quite useful for situations such as controlling uncertain physical systems modeled as ODEs.

cs.LG↗

arXiv2025

Accelerating Spectral Clustering on Quantum and Analog Platforms

We introduce a novel hybrid quantum-analog algorithm to perform graph clustering that exploits connections between the evolution of dynamical systems on graphs and the underlying graph spectra. This approach constitutes a new class of algorithms that combine emerging quantum and analog platforms to accelerate computations. Our hybrid algorithm is equivalent to spectral clustering and significantly reduces the computational complexity from $O(N^3)$ to $O(N)$, where $N$ is the number of nodes in the graph. We achieve this speedup by circumventing the need for explicit eigendecomposition of the normalized graph Laplacian matrix, which dominates the classical complexity, and instead leveraging quantum evolution of the Schrödinger equation followed by efficient analog computation for the dynamic mode decomposition (DMD) step. Specifically, while classical spectral clustering requires $O(N^3)$ operations to perform eigendecomposition, our method exploits the natural quantum evolution of states according to the graph Laplacian Hamiltonian in linear time, combined with the linear scaling for DMD that leverages efficient matrix-vector multiplications on analog hardware. We prove and demonstrate that this hybrid approach can extract the eigenvalues and scaled eigenvectors of the normalized graph Laplacian by evolving Schrödinger dynamics on quantum computers followed by DMD computations on analog devices, providing a significant computational advantage for large-scale graph clustering problems. Our demonstrations can be reproduced using our code that has been released at https://github.com/XingziXu/quantum-analog-clustering.

cs.DS↗

arXiv2024

On the Emergence of Ergodic Dynamics in Unique Games

The Unique Games Conjecture (UGC) constitutes a highly dynamic subarea within computational complexity theory, intricately linked to the outstanding P versus NP problem. Despite multiple insightful results in the past few years, a proof for the conjecture remains elusive. In this work, we construct a novel dynamical systems-based approach for studying unique games and, more generally, the field of computational complexity. We propose a family of dynamical systems whose equilibria correspond to solutions of unique games and prove that unsatisfiable instances lead to ergodic dynamics. Moreover, as the instance hardness increases, the weight of the invariant measure in the vicinity of the optimal assignments scales polynomially, sub-exponentially, or exponentially depending on the value gap. We numerically reproduce a previously hypothesized hardness plot associated with the UGC. Our results indicate that the UGC is likely true, subject to our proposed conjectures that link dynamical systems theory with computational complexity.

math.DS↗

arXiv2024

Non-Markovian Quantum Control via Model Maximum Likelihood Estimation and Reinforcement Learning

Reinforcement Learning (RL) techniques have been increasingly applied in optimizing control systems. However, their application in quantum systems is hampered by the challenge of performing closed-loop control due to the difficulty in measuring these systems. This often leads to reliance on assumed models, introducing model bias, a problem that is exacerbated in open quantum dynamics where Markovian approximations are not valid. To address these challenges, we propose a novel approach that incorporates the non-Markovian nature of the environment into a low-dimensional effective reservoir. By initially employing a series of measurements as a 'dataset', we utilize machine learning techniques to learn the effective quantum dynamics more efficiently than traditional tomographic methods. Our methodology aims to demonstrates that by integrating reinforcement learning with model learning, it is possible to devise control policies and models that can counteract decoherence in a spin-boson system. This approach may not only mitigates the issues of model bias but also provides a more accurate representation of quantum dynamics, paving the way for more effective quantum control strategies.

quant-ph↗

arXiv2023

math-PVS: A Large Language Model Framework to Map Scientific Publications to PVS Theories

As artificial intelligence (AI) gains greater adoption in a wide variety of applications, it has immense potential to contribute to mathematical discovery, by guiding conjecture generation, constructing counterexamples, assisting in formalizing mathematics, and discovering connections between different mathematical areas, to name a few. While prior work has leveraged computers for exhaustive mathematical proof search, recent efforts based on large language models (LLMs) aspire to position computing platforms as co-contributors in the mathematical research process. Despite their current limitations in logic and mathematical tasks, there is growing interest in melding theorem proving systems with foundation models. This work investigates the applicability of LLMs in formalizing advanced mathematical concepts and proposes a framework that can critically review and check mathematical reasoning in research papers. Given the noted reasoning shortcomings of LLMs, our approach synergizes the capabilities of proof assistants, specifically PVS, with LLMs, enabling a bridge between textual descriptions in academic papers and formal specifications in PVS. By harnessing the PVS environment, coupled with data ingestion and conversion mechanisms, we envision an automated process, called \emph{math-PVS}, to extract and formalize mathematical theorems from research papers, offering an innovative tool for academic review and discovery.

cs.AI↗

arXiv2023

Efficient Quantum Algorithms for Nonlinear Stochastic Dynamical Systems

In this paper, we propose efficient quantum algorithms for solving nonlinear stochastic differential equations (SDE) via the associated Fokker-Planck equation (FPE). We discretize the FPE in space and time using two well-known numerical schemes, namely Chang-Cooper and implicit finite difference. We then compute the solution of the resulting system of linear equations using the quantum linear systems algorithm. We present detailed error and complexity analyses for both these schemes and demonstrate that our proposed algorithms, under certain conditions, provably compute the solution to the FPE within prescribed $ε$ error bounds with polynomial dependence on state dimension $d$. Classical numerical methods scale exponentially with dimension, thus, our approach, under the aforementioned conditions, provides an \emph{exponential speed-up} over traditional approaches.

math.DS↗

arXiv2023

An efficient quantum algorithm for simulating polynomial differential equations

We present an efficient quantum algorithm to simulate nonlinear differential equations with polynomial vector fields of arbitrary degree on quantum platforms. Models of physical systems that are governed by ordinary differential equations (ODEs) or partial differential equation (PDEs) can be challenging to solve on classical computers due to high dimensionality, stiffness, nonlinearities, and sensitive dependence to initial conditions. For sparse $n$-dimensional linear ODEs, quantum algorithms have been developed which can produce a quantum state proportional to the solution in poly(log(nx)) time using the quantum linear systems algorithm (QLSA). Recently, this framework was extended to systems of nonlinear ODEs with quadratic polynomial vector fields by applying Carleman linearization that enables the embedding of the quadratic system into an approximate linear form. A detailed complexity analysis was conducted which showed significant computational advantage under certain conditions. We present an extension of this algorithm to deal with systems of nonlinear ODEs with $k$-th degree polynomial vector fields for arbitrary (finite) values of $k$. The steps involve: 1) mapping the $k$-th degree polynomial ODE to a higher dimensional quadratic polynomial ODE; 2) applying Carleman linearization to transform the quadratic ODE to an infinite-dimensional system of linear ODEs; 3) truncating and discretizing the linear ODE and solving using the forward Euler method and QLSA. Alternatively, one could apply Carleman linearization directly to the $k$-th degree polynomial ODE, resulting in a system of infinite-dimensional linear ODEs, and then apply step 3. This solution route can be computationally more efficient. We present detailed complexity analysis of the proposed algorithms, prove polynomial scaling of runtime on $k$ and demonstrate the framework on an example.

math.DS↗

arXiv2022

A Dynamic Mode Decomposition Approach for Decentralized Spectral Clustering of Graphs

We propose a novel robust decentralized graph clustering algorithm that is provably equivalent to the popular spectral clustering approach. Our proposed method uses the existing wave equation clustering algorithm that is based on propagating waves through the graph. However, instead of using a fast Fourier transform (FFT) computation at every node, our proposed approach exploits the Koopman operator framework. Specifically, we show that propagating waves in the graph followed by a local dynamic mode decomposition (DMD) computation at every node is capable of retrieving the eigenvalues and the local eigenvector components of the graph Laplacian, thereby providing local cluster assignments for all nodes. We demonstrate that the DMD computation is more robust than the existing FFT based approach and requires 20 times fewer steps of the wave equation to accurately recover the clustering information and reduces the relative error by orders of magnitude. We demonstrate the decentralized approach on a range of graph clustering problems.

cs.LG↗

arXiv2022

A Koopman framework for rare event simulation in stochastic differential equations

We exploit the relationship between the stochastic Koopman operator and the Kolmogorov backward equation to construct importance sampling schemes for stochastic differential equations. Specifically, we propose using eigenfunctions of the stochastic Koopman operator to approximate the Doob transform for an observable of interest (e.g., associated with a rare event) which in turn yields an approximation of the corresponding zero-variance importance sampling estimator. Our approach is broadly applicable and systematic, treating non-normal systems, non-gradient systems, and systems with oscillatory dynamics or rank-deficient noise in a common framework. In nonlinear settings where the stochastic Koopman eigenfunctions cannot be derived analytically, we use dynamic mode decomposition (DMD) methods to compute them numerically, but the framework is agnostic to the particular numerical method employed. Numerical experiments demonstrate that even coarse approximations of a few eigenfunctions, where the latter are built from non-rare trajectories, can produce effective importance sampling schemes for rare events.

stat.CO↗

arXiv2021

Computing eigenfunctions of the multidimensional Ornstein-Uhlenbeck operator

We discuss approaches to computing eigenfunctions of the Ornstein--Uhlenbeck (OU) operator in more than two dimensions. While the spectrum of the OU operator and theoretical properties of its eigenfunctions have been well characterized in previous research, the practical computation of general eigenfunctions has not been resolved. We review special cases for which the eigenfunctions can be expressed exactly in terms of commonly used orthogonal polynomials. Then we present a tractable approach for computing the eigenfunctions in general cases and comment on its dimension dependence.

math.NA↗

arXiv2021

Talaria: A Framework for Simulation of Permissioned Blockchains for Logistics and Beyond

In this paper, we present Talaria, a novel permissioned blockchain simulator that supports numerous protocols and use cases, most notably in supply chain management. Talaria extends the capability of BlockSim, an existing blockchain simulator, to include permissioned blockchains and serves as a foundation for further private blockchain assessment. Talaria is designed with both practical Byzantine Fault Tolerance (pBFT) and simplified version of Proof-of-Authority consensus protocols, but can be revised to include other permissioned protocols within its modular framework. Moreover, Talaria is able to simulate different types of malicious authorities and a variable daily transaction load at each node. In using Talaria, business practitioners and policy planners have an opportunity to measure, evaluate, and adapt a range of blockchain solutions for commercial operations.

cs.CR↗