arXiv subjects

Dibyajyoti Nayak

Publications and source records attributed to Dibyajyoti Nayak.

5 recordsLinked to original sources

arXiv2026

Data-Efficient Time-Dependent PDE Surrogates: Graph Neural Simulators vs. Neural Operators

Developing accurate, data-efficient surrogate models is central to advancing AI for Science. Neural operators (NOs), which approximate mappings between infinite-dimensional function spaces using conventional neural architectures, have gained popularity as surrogates for systems driven by partial differential equations (PDEs). However, their reliance on large datasets and limited ability to generalize in low-data regimes hinder their practical utility. We argue that these limitations arise from their global processing of data, which fails to exploit the local, discretized structure of physical systems. To address this, we propose Graph Neural Simulators (GNS) as a principled surrogate modeling paradigm for time-dependent PDEs. GNS leverages message-passing combined with numerical time-stepping schemes to learn PDE dynamics by modeling the instantaneous time derivatives. This design mimics traditional numerical solvers, enabling stable long-horizon rollouts and strong inductive biases that enhance generalization. We rigorously evaluate GNS on five canonical PDE systems: (1) 2D scalar Burgers', (2) 2D coupled Burgers', (3) 2D Allen-Cahn, (4) 2D nonlinear shallow water equations, and (5) 2D flow past a cylinder, comparing against SOTA NOs including DeepONet, FNO, and GINO. Results demonstrate that GNS is markedly more data-efficient, achieving less than 1% relative L2 error using only 3% of available trajectories, and exhibits dramatically reduced error accumulation over time (82.5% lower autoregressive error than FNO, 99.9% lower than DeepONet). To choose the training data, we introduce a PCA combined with KMeans trajectory selection strategy. These findings provide compelling evidence that GNS, with its graph-based locality and solver-inspired design, is the most suitable and scalable surrogate modeling framework for AI-driven scientific discovery.

cs.LG↗

arXiv2026

ANCHOR: Error-Controlled Adaptive Numerical Correction for Neural Operator Time Marching

Numerical simulation of time-dependent partial differential equations (PDEs) is central to scientific and engineering applications, but high-fidelity solvers are often prohibitively expensive for long-horizon or time-critical settings. Neural operator (NO) surrogates offer fast inference across parametric and functional inputs; however, most autoregressive NO frameworks remain vulnerable to compounding errors, and ensemble-averaged metrics provide limited guarantees for individual inference trajectories. In practice, error accumulation can become unacceptable beyond the training horizon, and existing methods lack mechanisms for online monitoring or correction. To address this gap, we propose ANCHOR (Adaptive Numerical Correction for High-fidelity Operator Rollouts), an online, instance-aware hybrid inference framework for stable long-horizon prediction of nonlinear, time-dependent PDEs. ANCHOR treats a pretrained NO as the primary inference engine and adaptively couples it with a classical numerical solver using a physics-informed, residual-based error estimator. Inspired by adaptive time-stepping in numerical analysis, ANCHOR monitors an exponential moving average (EMA) of the normalized PDE residual to detect accumulating error and trigger corrective solver interventions without requiring access to ground-truth solutions. We show that the EMA-based estimator correlates strongly with the true relative L2 error, enabling data-free, instance-aware error control during inference. Evaluations on six canonical PDEs: 1D and 2D Burgers', 2D Allen-Cahn, 2D Cahn-Hilliard, 2D Navier-Stokes, and 3D heat conduction, demonstrate that ANCHOR reliably bounds long-horizon error growth, stabilizes extrapolative rollouts, and significantly improves robustness over standalone neural operators, while remaining substantially more efficient than high-fidelity numerical solvers.

cs.LG↗

arXiv2026

TI-DeepONet: Learnable Time Integration for Stable Long-Term Extrapolation

Accurate temporal extrapolation remains a fundamental challenge for neural operators modeling dynamical systems, where predictions must extend far beyond the training horizon. Conventional DeepONet approaches rely on two limited paradigms: fixed-horizon rollouts, which predict full spatiotemporal solutions while ignoring temporal causality, and autoregressive schemes, which accumulate errors through sequential prediction. We introduce TI-DeepONet, a framework that integrates neural operators with adaptive numerical time-stepping to preserve the Markovian structure of dynamical systems while mitigating long-term error growth. Our method shifts the learning objective from direct state prediction to approximating instantaneous time-derivative fields, which are then integrated using standard numerical solvers. This naturally enables continuous-time prediction and allows the use of higher-order integrators at inference than those used in training, improving both efficiency and accuracy. We further propose TI(L)-DeepONet, which incorporates learnable coefficients for intermediate stages in multi-stage integration, adapting to solution-specific dynamics and enhancing fidelity. Across six canonical PDEs featuring chaotic, dissipative, dispersive, and high-dimensional behavior, TI(L)-DeepONet maeginally outperforms TI-DeepONet, and both achieve major reductions in relative L2 extrapolation error: about 96.3% compared to autoregressive methods and 83.6% compared to fixed-horizon approaches. Notably, both models maintain stable predictions over temporal domains nearly twice the training interval. This work establishes a physics-aware operator learning framework that bridges neural approximation with numerical analysis principles, addressing a key gap in long-term forecasting of complex physical systems.

cs.LG↗

arXiv2026

Physics-Informed Time-Integrated DeepONet: Temporal Tangent Space Operator Learning for High-Accuracy Inference

Accurately modeling and inferring solutions to time-dependent partial differential equations (PDEs) over extended horizons remains a core challenge in scientific machine learning. Traditional full rollout (FR) methods, which predict entire trajectories in one pass, often fail to capture the causal dependencies and generalize poorly outside the training time horizon. Autoregressive (AR) approaches, evolving the system step by step, suffer from error accumulation, limiting long-term accuracy. These shortcomings limit the long-term accuracy and reliability of both strategies. To address these issues, we introduce the Physics-Informed Time-Integrated Deep Operator Network (PITI-DeepONet), a dual-output architecture trained via physics-informed or hybrid physics- and data-driven objectives to ensure stable, accurate long-term evolution well beyond the training horizon. Instead of forecasting future states, the network learns the time-derivative operator from the current state, integrating it using classical time-stepping schemes to advance the solution in time. Additionally, the framework can leverage residual monitoring during inference to estimate prediction quality and detect when the system transitions outside the training domain. Applied to benchmark problems, PITI-DeepONet demonstrates enhanced accuracy and stability over extended inference time horizons when compared to traditional methods. Mean relative $\mathcal{L}_2$ errors reduced by 84\% (versus FR) and 79\% (versus AR) for 1D heat equation; by 87\% (versus FR) and 98\% (versus AR) for the 1D Burgers equation; by 42\% (versus FR) and 89\% (versus AR) for the 2D Allen-Cahn equation; and by 58\% (vs. FR) and 61\% (vs. AR) for the 1D Kuramoto-Sivashinsky equation. By moving beyond classic FR and AR schemes, PITI-DeepONet paves the way for more reliable, long-term integration of complex, time-dependent PDEs.

cs.LG↗

arXiv2023

A co-kurtosis PCA based dimensionality reduction with nonlinear reconstruction using neural networks

For turbulent reacting flows, identification of low-dimensional representations of the thermo-chemical state space is vitally important, primarily to significantly reduce the computational cost of device-scale simulations. Principal component analysis (PCA), and its variants, is a widely employed class of methods. Recently, an alternative technique that focuses on higher-order statistical interactions, co-kurtosis PCA (CoK-PCA), has been shown to effectively provide a low-dimensional representation by capturing the stiff chemical dynamics associated with spatiotemporally localized reaction zones. While its effectiveness has only been demonstrated based on a priori analysis with linear reconstruction, in this work, we employ nonlinear techniques to reconstruct the full thermo-chemical state and evaluate the efficacy of CoK-PCA compared to PCA. Specifically, we combine a CoK-PCA/PCA based dimensionality reduction (encoding) with an artificial neural network (ANN) based reconstruction (decoding) and examine a priori the reconstruction errors of the thermo-chemical state. In addition, we evaluate the errors in species production rates and heat release rates that are nonlinear functions of the reconstructed state as a measure of the overall accuracy of the dimensionality reduction technique. We employ four datasets to assess CoK-PCA/PCA coupled with ANN-based reconstruction: a homogeneous reactor for autoignition of an ethylene/air mixture that has conventional single-stage ignition kinetics, a dimethyl ether (DME)/air mixture which has two-stage ignition kinetics, a one-dimensional freely propagating premixed ethylene/air laminar flame, and a two-dimensional homogeneous charge compression ignition of ethanol. The analyses demonstrate the robustness of the CoK-PCA based low-dimensional manifold with ANN reconstruction in accurately capturing the data, specifically from the reaction zones.

physics.flu-dyn↗