Partial differential equations (PDEs) are fundamental to modeling complex and nonlinear physical phenomena, but their numerical solution often requires significant computational resources, particularly when a large number of forward full solution evaluations are necessary, such as in design, optimization, sensitivity analysis, and uncertainty quantification. Recent progress in operator learning has enabled surrogate models that efficiently predict full PDE solution fields; however, these models often struggle with accuracy and robustness when faced with highly nonlinear responses driven by sequential input functions. To address these challenges, we propose the Sequential Neural Operator Transformer (S-NOT), a architecture that combines gated recurrent units (GRUs) with the self-attention mechanism of transformers to address time-dependent,nonlinear PDEs. Unlike S-DeepONet (S-DON), which uses a dot product to merge encoded outputs from the branch and trunk sub-networks, S-NOT leverages attention to better capture intricate dependencies between sequential inputs and spatial query points. We benchmark S-NOT on three challenging datasets from real-world applications with plastic and thermo-viscoplastic highly nonlinear material responses: multiphysics steel solidification, a 3D lug specimen, and a dogbone specimen under temporal and path-dependent loadings. The results show that S-NOT consistently achieves a higher prediction accuracy than S-DON even for data outliers, demonstrating its accuracy and robustness for drastically accelerating computational frameworks in scientific and engineering applications.

Accurate reconstruction of latent environmental fields from sparse, indirect observations is a fundamental challenge across scientific domains, from atmospheric science and geophysics to public health and aerospace safety. Existing approaches typically rely on physics-based simulations or dense sensor networks; however, these methods are hampered by high computational cost, latency, and limited spatial coverage. Here we introduce the \textbf{Temporal Radiation Operator Network (TRON)}, a spatiotemporal neural operator architecture that infers continuous global scalar fields solely from sequences of sparse, non-uniform proxy measurements. Unlike recent prediction models that require dense, gridded inputs to predict system states, TRON tackles sparse-to-dense, cross-domain field reconstruction. It reconstructs the current global field in real time from sparse, temporally evolving sensor data, without access to any future observations or dense ground-truth fields. We demonstrate this approach on global cosmic radiation dose mapping: TRON, trained on daily reference fields spanning 2001 to 2023, generalizes across 65,341 spatial locations with input sequences ranging from 7 to 90 days. It achieves sub-second inference with relative $L_2$ errors below 0.1\%, representing over a 58,000$\times$ speedup compared to physics-based estimators. Although showcased in the radiation application, TRON provides a domain-agnostic framework for continuous field reconstruction from sparse data, with broad applications in atmospheric modeling, geophysical hazard monitoring, and real-time environmental risk prediction.

Much of the physical world is inaccessible to direct measurement and can be reached only through the sparse traces it leaves in other observable variables. Here we introduce the Temporal Radiation Operator Network (TRON), a spatiotemporal neural operator that learns proxy-to-field mappings, reconstructing otherwise unobserved physical fields from sparse, indirect sensor histories. For cosmic-radiation mapping, TRON reconstructs sea-level effective dose at 65,341 locations from histories of 12 neutron monitors, achieving 0.096\% mean relative $L_2$ error against EXPACS-generated reference fields in a chronologically held-out year; batched GPU inference evaluates a global field in approximately $\sim$1~ms, compared with $\sim$57.6~s for EXPACS on one CPU thread. On measured weather networks, TRON reconstructs temperature at held-out stations from thermometer or barometer histories, outperforming interpolation, calendar-based and hypsometric baselines. These results establish proxy-to-field reconstruction as a general route to recovering spatially distributed physical information beyond the direct reach of sparse sensor networks.

Machine-learning-based surrogate models offer significant computational efficiency and faster simulations compared to traditional numerical methods, especially for problems requiring repeated evaluations of partial differential equations. This work introduces the Geometry-Informed Neural Operator Transformer (GINOT), which integrates the transformer architecture with the neural operator framework to enable forward predictions on arbitrary geometries. GINOT employs a sampling and grouping strategy together with an attention mechanism to encode surface point clouds that are unordered, exhibit non-uniform point densities, and contain varying numbers of points for different geometries. The geometry information is seamlessly integrated with query points in the solution decoder through the attention mechanism. The performance of GINOT is validated on multiple challenging datasets, showcasing its high accuracy and strong generalization capabilities for complex and arbitrary 2D and 3D geometries.

The inverse design of metamaterial architectures presents a significant challenge, particularly for nonlinear mechanical properties involving large deformations, buckling, contact, and plasticity. Traditional methods, such as gradient-based optimization, and recent generative deep-learning approaches often rely on binary pixel-based representations, which introduce jagged edges that hinder finite element (FE) simulations and 3D printing. To overcome these challenges, we propose an inverse design framework that utilizes a signed distance function (SDF) representation combined with a conditional diffusion model. The SDF provides a smooth boundary representation, eliminating the need for post-processing and ensuring compatibility with FE simulations and manufacturing methods. A classifier-free guided diffusion model is trained to generate SDFs conditioned on target macroscopic stress-strain curves, enabling efficient one-shot design synthesis. To assess the mechanical response of the generated designs, we introduce a forward prediction model based on Neural Operator Transformers (NOT), which accurately predicts homogenized stress-strain curves and local solution fields for arbitrary geometries with irregular query meshes. This approach enables a closed-loop process for general metamaterial design, offering a pathway for the development of advanced functional materials.

Rapid reaction-thermal diffusion during frontal polymerization (FP) with variations in initial and boundary conditions destabilizes the planar mode of front propagation, leading to spatially varying complex hierarchical patterns in polymeric materials. Although modern reaction-diffusion models can predict the patterns resulting from unstable FP, the inverse design of patterns, which aims to retrieve process conditions that produce a desired pattern, remains an open challenge due to the nonunique and nonintuitive mapping between process conditions and patterns. In this work, we propose a novel probabilistic generative model named univariate conditional variational autoencoder (UcVAE) for the inverse design of hierarchical patterns in FP-based manufacturing. Unlike the cVAE, which encodes both the design space and the design target, the UcVAE encodes only the design space. In the encoder of the UcVAE, the number of training parameters is significantly reduced compared to the cVAE, resulting in a shorter training time while maintaining comparable performance. Given desired pattern images, the trained UcVAE can generate multiple process condition solutions that produce high-fidelity hierarchical patterns.

Under some initial and boundary conditions, the rapid reaction-thermal diffusion process taking place during frontal polymerization (FP) destabilizes the planar mode of front propagation, leading to spatially varying, complex hierarchical patterns in thermoset polymeric materials. Although modern reaction-diffusion models can predict the patterns resulting from unstable FP, the inverse design of patterns, which aims to retrieve process conditions that produce a desired pattern, remains an open challenge due to the non-unique and non-intuitive mapping between process conditions and manufactured patterns. In this work, we propose a probabilistic generative model named univariate conditional variational autoencoder (UcVAE) for the inverse design of hierarchical patterns in FP-based manufacturing. Unlike the cVAE, which encodes both the design space and the design target, the UcVAE encodes only the design space. In the encoder of the UcVAE, the number of training parameters is significantly reduced compared to the cVAE, resulting in a shorter training time while maintaining comparable performance. Given desired pattern images, the trained UcVAE can generate multiple process condition solutions that produce high-fidelity hierarchical patterns.

Effective real-time monitoring technique is crucial for detecting material degradation and maintaining the structural integrity of nuclear systems to ensure both safety and operational efficiency. Traditional physical sensor systems face limitations such as installation challenges, high costs, and difficulties in measuring critical parameters in hard-to-reach or harsh environments, often resulting in incomplete data coverage. Machine learning-driven virtual sensors offer a promising solution by enhancing physical sensor capabilities to monitor critical degradation indicators like pressure, velocity, and turbulence. However, conventional machine learning models struggle with real-time monitoring due to the high-dimensional nature of reactor data and the need for frequent retraining. This paper explores the use of Deep Operator Networks (DeepONet) within a digital twin (DT) framework to predict key thermal-hydraulic parameters in the hot leg of an AP-1000 Pressurized Water Reactor (PWR). In this study, DeepONet is trained with different operational conditions, which relaxes the requirement of continuous retraining, making it suitable for online and real-time prediction components for DT. Our results show that DeepONet achieves accurate predictions with low mean squared error and relative L2 error and can make predictions on unknown data 160,000 times faster than traditional finite element (FE) simulations. This speed and accuracy make DeepONet a powerful tool for tracking conditions that contribute to material degradation in real-time, enhancing reactor safety and longevity.

Effective real-time monitoring technique is crucial for detecting material degradation and maintaining the structural integrity of nuclear systems to ensure both safety and operational efficiency. Traditional physical sensor systems face limitations such as installation challenges, high costs, and difficulties in measuring critical parameters in hard-to-reach or harsh environments, often resulting in incomplete data coverage. Machine learning-driven virtual sensors offer a promising solution by enhancing physical sensor capabilities to monitor critical degradation indicators like pressure, velocity, and turbulence. However, conventional machine learning models struggle with real-time monitoring due to the high-dimensional nature of reactor data and the need for frequent retraining. This paper explores the use of Deep Operator Networks (DeepONet) within a digital twin (DT) framework to predict key thermal-hydraulic parameters in the hot leg of an AP-1000 Pressurized Water Reactor (PWR). In this study, DeepONet is trained with different operational conditions, which relaxes the requirement of continuous retraining, making it suitable for online and real-time prediction components for DT. Our results show that DeepONet achieves accurate predictions with low mean squared error and relative L2 error and can make predictions on unknown data 160,000 times faster than traditional finite element (FE) simulations. This speed and accuracy make DeepONet a powerful tool for tracking conditions that contribute to material degradation in real-time, enhancing reactor safety and longevity.

Real-time monitoring is a foundation of nuclear digital twin technology, crucial for detecting material degradation and maintaining nuclear system integrity. Traditional physical sensor systems face limitations, particularly in measuring critical parameters in hard-to-reach or harsh environments, often resulting in incomplete data coverage. Machine learning-driven virtual sensors offer a transformative solution by complementing physical sensors in monitoring critical degradation indicators. This paper introduces the use of Deep Operator Networks (DeepONet) to predict key thermal-hydraulic parameters in the hot leg of pressurized water reactor. DeepONet acts as a virtual sensor, mapping operational inputs to spatially distributed system behaviors without requiring frequent retraining. Our results show that DeepONet achieves low mean squared and Relative L2 error, making predictions 1400 times faster than traditional CFD simulations. These characteristics enable DeepONet to function as a real-time virtual sensor, synchronizing with the physical system to track degradation conditions and provide insights within the digital twin framework for nuclear systems.