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 for arbitrary geometries. GINOT encodes the surface points cloud of a geometry using a sampling and grouping mechanism combined with an attention mechanism, ensuring invariance to point order and padding while maintaining robustness to variations in point density. 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.

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.

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.

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.

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.

Metamaterials, synthetic materials with customized properties, have emerged as a promising field due to advancements in additive manufacturing. These materials derive unique mechanical properties from their internal lattice structures, which are often composed of multiple materials that repeat geometric patterns. While traditional inverse design approaches have shown potential, they struggle to map nonlinear material behavior to multiple possible structural configurations. This paper presents a novel framework leveraging video diffusion models, a type of generative artificial Intelligence (AI), for inverse multi-material design based on nonlinear stress-strain responses. Our approach consists of two key components: (1) a fields generator using a video diffusion model to create solution fields based on target nonlinear stress-strain responses, and (2) a structure identifier employing two UNet models to determine the corresponding multi-material 2D design. By incorporating multiple materials, plasticity, and large deformation, our innovative design method allows for enhanced control over the highly nonlinear mechanical behavior of metamaterials commonly seen in real-world applications. It offers a promising solution for generating next-generation metamaterials with finely tuned mechanical characteristics.

Metamaterials, synthetic materials with customized properties, have emerged as a promising field due to advancements in additive manufacturing. These materials derive unique mechanical properties from their internal lattice structures, which are often composed of multiple materials that repeat geometric patterns. While traditional inverse design approaches have shown potential, they struggle to map nonlinear material behavior to multiple possible structural configurations. This paper presents a novel framework leveraging video diffusion models, a type of generative artificial Intelligence (AI), for inverse multi-material design based on nonlinear stress-strain responses. Our approach consists of two key components: (1) a fields generator using a video diffusion model to create solution fields based on target nonlinear stress-strain responses, and (2) a structure identifier employing two UNet models to determine the corresponding multi-material 2D design. By incorporating multiple materials, plasticity, and large deformation, our innovative design method allows for enhanced control over the highly nonlinear mechanical behavior of metamaterials commonly seen in real-world applications. It offers a promising solution for generating next-generation metamaterials with finely tuned mechanical characteristics.

Data-driven models that act as surrogates for computationally costly 3D topology optimization techniques are very popular because they help alleviate multiple time-consuming 3D finite element analyses during optimization. In this study, one such 3D CNN-based surrogate model for the topology optimization of Schoen’s gyroid triply periodic minimal surface unit cell is investigated. Gyroid-like unit cells are designed using a voxel algorithm and homogenization-based topology optimization codes in MATLAB. A few such optimization data are used as input–output for supervised learning of the topology-optimization process via the 3D CNN model in Python code. These models could then be used to instantaneously predict the optimized unit cell geometry for any topology parameters. The high accuracy of the model was demonstrated by a low mean square error metric and a high Dice coefficient metric. The model has the major disadvantage of running numerous costly topology optimization runs but has the advantages that the trained model can be reused for different cases of TO and that the methodology of the accelerated design of 3D metamaterials can be extended for designing any complex, computationally costly problems of metamaterials with multi-objective properties or multiscale applications. The main purpose of this paper is to provide the complete associated MATLAB and PYTHON codes for optimizing the topology of any cellular structure and predicting new topologies using deep learning for educational purposes.