Hamidreza Eivazi

Multiscale problems are ubiquitous in physics. Numerical simulations of such problems by solving partial differential equations at high resolution are computationally too expensive for many-query scenarios, such as uncertainty quantification, remeshing applications, and topology optimization. This limitation has motivated data-driven surrogate models that replace microscale computations with black-box mappings between macroscale quantities. Although these approaches can provide substantial speedups, they often struggle to incorporate microscale physical constraints such as the balance of linear momentum. In this work, we propose the Equilibrium Neural Operator (EquiNO), a physics-informed PDE surrogate in which equilibrium is enforced by construction. EquiNO projects the solution onto divergence-free basis functions obtained through proper orthogonal decomposition, satisfying equilibrium without penalty terms or multi-objective loss functions. We compare EquiNO with variational physics-informed neural and operator networks, as well as purely data-driven operator-learning baselines. The proposed finite element-operator learning framework is applied to quasi-static solid-mechanics problems and yields accurate solutions even with restricted training datasets. EquiNO achieves speedup factors exceeding 8000-fold compared with traditional FE2 simulations while providing a robust and physically consistent surrogate model.

Battery degradation remains a critical challenge in the pursuit of green technologies and sustainable energy solutions. Despite significant research efforts, predicting battery capacity loss accurately remains a formidable task due to its complex nature, influenced by both aging and cycling behaviors. To address this challenge, we introduce a novel general-purpose model for battery degradation prediction and synthesis, DiffBatt. Leveraging an innovative combination of conditional and unconditional diffusion models with classifier-free guidance and transformer architecture, DiffBatt achieves high expressivity and scalability. DiffBatt operates as a probabilistic model to capture uncertainty in aging behaviors and a generative model to simulate battery degradation. The performance of the model excels in prediction tasks while also enabling the generation of synthetic degradation curves, facilitating enhanced model training by data augmentation. In the remaining useful life prediction task, DiffBatt provides accurate results with a mean RMSE of 196 cycles across all datasets, outperforming all other models and demonstrating superior generalizability. This work represents an important step towards developing foundational models for battery degradation.

Physics-informed neural networks (PINNs) are successful machine-learning methods for the solution and identification of partial differential equations. We employ PINNs for solving the Reynolds-averaged Navier–Stokes equations for incompressible turbulent flows without any specific model or assumption for turbulence and by taking only the data on the domain boundaries. We first show the applicability of PINNs for solving the Navier–Stokes equations for laminar flows by solving the Falkner–Skan boundary layer. We then apply PINNs for the simulation of four turbulent-flow cases, i.e., zero-pressure-gradient boundary layer, adverse-pressure-gradient boundary layer, and turbulent flows over a NACA4412 airfoil and the periodic hill. Our results show the excellent applicability of PINNs for laminar flows with strong pressure gradients, where predictions with less than 1\% error can be obtained. For turbulent flows, we also obtain very good accuracy on simulation results even for the Reynolds-stress components.

Modal-decomposition techniques are computational frameworks based on data aimed at identifying a low-dimensional space for capturing dominant flow features: the so-called modes. We propose a deep probabilistic-neural-network architecture for learning a minimal and near-orthogonal set of non-linear modes from high-fidelity turbulent-flow data useful for flow analysis, reduced-order modeling and flow control. Our approach is based on β-variational autoencoders (β-VAEs) and convolutional neural networks (CNNs), which enable extracting non-linear modes from multi-scale turbulent flows while encouraging the learning of independent latent variables and penalizing the size of the latent vector. Moreover, we introduce an algorithm for ordering VAE-based modes with respect to their contribution to the reconstruction. We apply this method for non-linear mode decomposition of the turbulent flow through a simplified urban environment, where the flow-field data is obtained based on well-resolved large-eddy simulations (LESs). We demonstrate that by constraining the shape of the latent space, it is possible to motivate the orthogonality and extract a set of parsimonious modes sufficient for high-quality reconstruction. Our results show the excellent performance of the method in the reconstruction against linear-theory-based decompositions, where the energy percentage captured by the proposed method from five modes is equal to 87.36% against 32.41% of the POD. Moreover, we compare our method with available AE-based models. We show the ability of our approach in the extraction of near-orthogonal modes with the determinant of the correlation matrix equal to 0.99, which may lead to interpretability.

Operator learning provides methods to approximate mappings between infinite-dimensional function spaces. Deep operator networks (DeepONets) are a notable architecture in this field. Recently, an extension of DeepONet based on model reduction and neural networks, proper orthogonal decomposition (POD)-DeepONet, has been able to outperform other architectures in terms of accuracy for several benchmark tests. We extend this idea towards nonlinear model order reduction by proposing an efficient framework that combines neural networks with kernel principal component analysis (KPCA) for operator learning. Our results demonstrate the superior performance of KPCA-DeepONet over POD-DeepONet.