Aniruddha Bora

Somdatta Goswami, Aniruddha Bora, Yue Yu et al.

Standard neural networks can approximate general nonlinear operators, represented either explicitly by a combination of mathematical operators, e.g., in an advection-diffusion-reaction partial differential equation, or simply as a black box, e.g., a system-of-systems. The first neural operator was the Deep Operator Network (DeepONet), proposed in 2019 based on rigorous approximation theory. Since then, a few other less general operators have been published, e.g., based on graph neural networks or Fourier transforms. For black box systems, training of neural operators is data-driven only but if the governing equations are known they can be incorporated into the loss function during training to develop physics-informed neural operators. Neural operators can be used as surrogates in design problems, uncertainty quantification, autonomous systems, and almost in any application requiring real-time inference. Moreover, independently pre-trained DeepONets can be used as components of a complex multi-physics system by coupling them together with relatively light training. Here, we present a review of DeepONet, the Fourier neural operator, and the graph neural operator, as well as appropriate extensions with feature expansions, and highlight their usefulness in diverse applications in computational mechanics, including porous media, fluid mechanics, and solid mechanics.

Aniruddha Bora, Khemraj Shukla, Shixuan Zhang et al.

Numerical simulation for climate modeling resolving all important scales is a computationally taxing process. Therefore, to circumvent this issue a low resolution simulation is performed, which is subsequently corrected for bias using reanalyzed data (ERA5), known as nudging correction. The existing implementation for nudging correction uses a relaxation based method for the algebraic difference between low resolution and ERA5 data. In this study, we replace the bias correction process with a surrogate model based on the Deep Operator Network (DeepONet). DeepONet (Deep Operator Neural Network) learns the mapping from the state before nudging (a functional) to the nudging tendency (another functional). The nudging tendency is a very high dimensional data albeit having many low energy modes. Therefore, the DeepoNet is combined with a convolution based auto-encoder-decoder (AED) architecture in order to learn the nudging tendency in a lower dimensional latent space efficiently. The accuracy of the DeepONet model is tested against the nudging tendency obtained from the E3SMv2 (Energy Exascale Earth System Model) and shows good agreement. The overarching goal of this work is to deploy the DeepONet model in an online setting and replace the nudging module in the E3SM loop for better efficiency and accuracy.

Alan John Varghese, Aniruddha Bora, Mengjia Xu et al.

Dynamic graph embedding has emerged as a very effective technique for addressing diverse temporal graph analytic tasks (i.e., link prediction, node classification, recommender systems, anomaly detection, and graph generation) in various applications. Such temporal graphs exhibit heterogeneous transient dynamics, varying time intervals, and highly evolving node features throughout their evolution. Hence, incorporating long-range dependencies from the historical graph context plays a crucial role in accurately learning their temporal dynamics. In this paper, we develop a graph embedding model with uncertainty quantification, TransformerG2G, by exploiting the advanced transformer encoder to first learn intermediate node representations from its current state ($t$) and previous context (over timestamps [$t-1, t-l$], $l$ is the length of context). Moreover, we employ two projection layers to generate lower-dimensional multivariate Gaussian distributions as each node's latent embedding at timestamp $t$. We consider diverse benchmarks with varying levels of ``novelty" as measured by the TEA (Temporal Edge Appearance) plots. Our experiments demonstrate that the proposed TransformerG2G model outperforms conventional multi-step methods and our prior work (DynG2G) in terms of both link prediction accuracy and computational efficiency, especially for high degree of novelty. Furthermore, the learned time-dependent attention weights across multiple graph snapshots reveal the development of an automatic adaptive time stepping enabled by the transformer. Importantly, by examining the attention weights, we can uncover temporal dependencies, identify influential elements, and gain insights into the complex interactions within the graph structure. For example, we identified a strong correlation between attention weights and node degree at the various stages of the graph topology evolution.

Vivek Oommen, Aniruddha Bora, Zhen Zhang et al.

We integrate neural operators with diffusion models to address the spectral limitations of neural operators in surrogate modeling of turbulent flows. While neural operators offer computational efficiency, they exhibit deficiencies in capturing high-frequency flow dynamics, resulting in overly smooth approximations. To overcome this, we condition diffusion models on neural operators to enhance the resolution of turbulent structures. Our approach is validated for different neural operators on diverse datasets, including a high Reynolds number jet flow simulation and experimental Schlieren velocimetry. The proposed method significantly improves the alignment of predicted energy spectra with true distributions compared to neural operators alone. This enables the diffusion models to stabilize longer forecasts through diffusion-corrected autoregressive rollouts, as we demonstrate in this work. Additionally, proper orthogonal decomposition analysis demonstrates enhanced spectral fidelity in space-time. This work establishes a new paradigm for combining generative models with neural operators to advance surrogate modeling of turbulent systems, and it can be used in other scientific applications that involve microstructure and high-frequency content. See our project page: vivekoommen.github.io/NO_DM

Aniruddha Bora, Shixuan Zhang, Khemraj Shukla et al.

Coarse resolution, imperfect parameterizations, and uncertain initial states and forcings limit Earth-system model (ESM) predictions. Traditional bias correction via data assimilation improves constrained simulations but offers limited benefit once models run freely. We introduce an operator-learning framework that maps instantaneous model states to bias-correction tendencies and applies them online during integration. Building on a U-Net backbone, we develop two operator architectures Inception U-Net (IUNet) and a multi-scale network (M\&M) that combine diverse upsampling and receptive fields to capture multiscale nonlinear features under Energy Exascale Earth System Model (E3SM) runtime constraints. Trained on two years E3SM simulations nudged toward ERA5 reanalysis, the operators generalize across height levels and seasons. Both architectures outperform standard U-Net baselines in offline tests, indicating that functional richness rather than parameter count drives performance. In online hybrid E3SM runs, M\&M delivers the most consistent bias reductions across variables and vertical levels. The ML-augmented configurations remain stable and computationally feasible in multi-year simulations, providing a practical pathway for scalable hybrid modeling. Our framework emphasizes long-term stability, portability, and cadence-limited updates, demonstrating the utility of expressive ML operators for learning structured, cross-scale relationships and retrofitting legacy ESMs.

Aniruddha Bora, Julie Chalfant, Chryssostomos Chryssostomidis

International shipping produces approximately 3% of global greenhouse gas emissions, yet voyage routing remains dominated by heuristic methods. We present PIER (Physics-Informed, Energy-efficient, Risk-aware routing), an offline reinforcement learning framework that learns fuel-efficient, safety-aware routing policies from physics-calibrated environments grounded in historical vessel tracking data and ocean reanalysis products, requiring no online simulator. Validated on one full year (2023) of AIS data across seven Gulf of Mexico routes (840 episodes per method), PIER reduces mean CO2 emissions by 10% relative to great-circle routing. However, PIER's primary contribution is eliminating catastrophic fuel waste: great-circle routing incurs extreme fuel consumption (>1.5x median) in 4.8% of voyages; PIER reduces this to 0.5%, a 9-fold reduction. Per-voyage fuel variance is 3.5x lower (p<0.001), with bootstrap 95% CI for mean savings [2.9%, 15.7%]. Partial validation against observed AIS vessel behavior confirms consistency with the fastest real transits while exhibiting 23.1x lower variance. Crucially, PIER is forecast-independent: unlike A* path optimization whose wave protection degrades 4.5x under realistic forecast uncertainty, PIER maintains constant performance using only local observations. The framework combines physics-informed state construction, demonstration-augmented offline data, and a decoupled post-hoc safety shield, an architecture that transfers to wildfire evacuation, aircraft trajectory optimization, and autonomous navigation in unmapped terrain.

Aniruddha Bora, Isabel K. Alvarez, Julie Chalfant et al.

In this work, we present a methodology using Physics Informed Neural Networks (PINNs) to determine the required velocity of a coolant, given inlet and outlet temperatures for a given heat flux in a multilayered metal-oxide-semiconductor field-effect transistor (MOSFET). MOSFETs are integral components of Power Electronic Building Blocks (PEBBs) and experiences the majority of the thermal load. Effective cooling of MOSFETs is therefore essential to prevent overheating and potential burnout. Determining the required velocity for the purpose of effective cooling is of importance but is an ill-posed inverse problem and difficult to solve using traditional methods. MOSFET consists of multiple layers with different thermal conductivities, including aluminum, pyrolytic graphite sheets (PGS), and stainless steel pipes containing flowing water. We propose an algorithm that employs sequential training of the MOSFET layers in PINNs. Mathematically, the sequential training method decouples the optimization of each layer by treating the parameters of other layers as constants during its training phase. This reduces the dimensionality of the optimization landscape, making it easier to find the global minimum for each layer's parameters and avoid poor local minima. Convergence of the PINNs solution to the analytical solution is theoretically analyzed. Finally we show the prediction of our proposed methodology to be in good agreement with experimental results.

Siavash Khodakarami, Vivek Oommen, Aniruddha Bora et al.

Neural operators have emerged as powerful surrogates for modeling complex physical problems. However, they suffer from spectral bias making them oblivious to high-frequency modes, which are present in multiscale physical systems. Therefore, they tend to produce over-smoothed solutions, which is particularly problematic in modeling turbulence and for systems with intricate patterns and sharp gradients such as multi-phase flow systems. In this work, we introduce a new approach named high-frequency scaling (HFS) to mitigate spectral bias in convolutional-based neural operators. By integrating HFS with proper variants of UNet neural operators, we demonstrate a higher prediction accuracy by mitigating spectral bias in single and two-phase flow problems. Unlike Fourier-based techniques, HFS is directly applied to the latent space, thus eliminating the computational cost associated with the Fourier transform. Additionally, we investigate alternative spectral bias mitigation through diffusion models conditioned on neural operators. While the diffusion model integrated with the standard neural operator may still suffer from significant errors, these errors are substantially reduced when the diffusion model is integrated with a HFS-enhanced neural operator.

Vivek Oommen, Siavash Khodakarami, Aniruddha Bora et al.

Neural operators are promising surrogates for dynamical systems but when trained with standard L2 losses they tend to oversmooth fine-scale turbulent structures. Here, we show that combining operator learning with generative modeling overcomes this limitation. We consider three practical turbulent-flow challenges where conventional neural operators fail: spatio-temporal super-resolution, forecasting, and sparse flow reconstruction. For Schlieren jet super-resolution, an adversarially trained neural operator (adv-NO) reduces the energy-spectrum error by 15x while preserving sharp gradients at neural operator-like inference cost. For 3D homogeneous isotropic turbulence, adv-NO trained on only 160 timesteps from a single trajectory forecasts accurately for five eddy-turnover times and offers 114x wall-clock speed-up at inference than the baseline diffusion-based forecasters, enabling near-real-time rollouts. For reconstructing cylinder wake flows from highly sparse Particle Tracking Velocimetry-like inputs, a conditional generative model infers full 3D velocity and pressure fields with correct phase alignment and statistics. These advances enable accurate reconstruction and forecasting at low compute cost, bringing near-real-time analysis and control within reach in experimental and computational fluid mechanics. See our project page: https://vivekoommen.github.io/Gen4Turb/

Ramona Rubini, Siavash Khodakarami, Aniruddha Bora et al.

Accurate time-series forecasting for complex physical systems is the backbone of modern industrial monitoring and control. While deep learning models excel at capturing complex dynamics, currently, their deployment is limited due to physical inconsistency and robustness, hence constraining their reliability in regulated environments. We introduce process-informed forecasting (PIF) models for temperature in pharmaceutical lyophilization. We investigate a wide range of models, from classical ones such as Autoregressive Integrated Moving Average Model (ARIMA) and Exponential Smoothing Model (ETS), to modern deep learning architectures, including Kolmogorov-Arnold Networks (KANs). We compare three different loss function formulations that integrate a process-informed trajectory prior: a fixed-weight loss, a dynamic uncertainty-based loss, and a Residual-Based Attention (RBA) mechanism. We evaluate all models not only for accuracy and physical consistency but also for robustness to sensor noise. Furthermore, we test the practical generalizability of the best model in a transfer learning scenario on a new process. Our results show that PIF models outperform their data-driven counterparts in terms of accuracy, physical plausibility and noise resilience. This work provides a roadmap for developing reliable and generalizable forecasting solutions for critical applications in the pharmaceutical manufacturing landscape.

Jiawen Wei, Aniruddha Bora, Vivek Oommen et al.

Extreme weather events are increasing in frequency and intensity due to climate change. This, in turn, is exacting a significant toll in communities worldwide. While prediction skills are increasing with advances in numerical weather prediction and artificial intelligence tools, extreme weather still present challenges. More specifically, identifying the precursors of such extreme weather events and how these precursors may evolve under climate change remain unclear. In this paper, we propose to use post-hoc interpretability methods to construct relevance weather maps that show the key extreme-weather precursors identified by deep learning models. We then compare this machine view with existing domain knowledge to understand whether deep learning models identified patterns in data that may enrich our understanding of extreme-weather precursors. We finally bin these relevant maps into different multi-year time periods to understand the role that climate change is having on these precursors. The experiments are carried out on Indochina heatwaves, but the methodology can be readily extended to other extreme weather events worldwide.