PDE Surrogate Modeling

Developing neural operators that accurately predict the behavior of systems governed by partial differential equations (PDEs) across unseen parameter regimes is crucial for robust generalization in scientific and engineering applications. In practical applications, variations in physical parameters induce distribution shifts between training and prediction regimes, making extrapolation a central challenge. As a result, the way parameters are incorporated into neural operator models plays a key role in their ability to generalize, particularly when state and parameter representations are entangled. In this work, we introduce the Late Fusion Neural Operator, an architecture that disentangles learning state dynamics from parameter effects, improving predictive performance both within and beyond the training distribution. Our approach combines neural operators for learning latent state representations with sparse regression to incorporate parameter information in a structured manner. Across four benchmark PDEs including advection, Burgers, and both 1D and 2D reaction-diffusion equations, the proposed method consistently outperforms Fourier Neural Operator and CAPE-FNO. Late Fusion Neural Operators achieve consistently the best performance in all experiments, with an average RMSE reduction of 72.9% in-domain and 71.8% out-domain compared to the second-best method. These results demonstrate strong generalization across both in-domain and out-domain parameter regimes.

Directed energy deposition (DED) produces complex thermo-mechanical responses that can lead to distortion and reduced dimensional accuracy of a manufactured part. Thermo-mechanical finite element simulations are widely used to estimate these effects, but their computational cost and the complexity of accurately capturing DED physics limit their use in design iteration and process optimization. This paper introduces FLARE (Field Prediction via Linear Affine Reconstruction in wEight-space), a data-efficient surrogate modeling framework for predicting post-cooling displacement fields in DED from geometric and process parameters. We develop a predefined-geometry DED simulation workflow using an open-source finite element framework and generate a dataset of simulations with varying geometry, laser power, and deposition velocity. Each simulation provides full-field displacement, stress, strain, and temperature data throughout the manufacturing process. FLARE encodes each simulation as an implicit neural field and regularizes the corresponding neural-network weights so that they follow the affine structure of the input parameter space. This enables prediction of unseen parameter combinations by reconstructing network weights through affine mixing of training examples. On this DED benchmark, the method shows improved accuracy compared to baseline methods in both in-distribution and extrapolation settings. Although the present study focuses on DED displacement prediction, the proposed affine weight-space reconstruction framework offers a promising approach for data-efficient surrogate modeling of physical fields.

We present PINNACLE, an open-source computational framework for physics-informed neural networks (PINNs) that integrates modern training strategies, multi-GPU acceleration, and hybrid quantum-classical architectures within a unified modular workflow. The framework enables systematic evaluation of PINN performance across benchmark problems including 1D hyperbolic conservation laws, incompressible flows, and electromagnetic wave propagation. It supports a range of architectural and training enhancements, including Fourier feature embeddings, random weight factorization, strict boundary condition enforcement, adaptive loss balancing, curriculum training, and second-order optimization strategies, with extensibility to additional methods. We provide a comprehensive benchmark study quantifying the impact of these methods on convergence, accuracy, and computational cost, and analyze distributed data parallel scaling in terms of runtime and memory efficiency. In addition, we extend the framework to hybrid quantum-classical PINNs and derive a formal estimate for circuit-evaluation complexity under parameter-shift differentiation. Results highlight the sensitivity of PINNs to architectural and training choices, confirm their high computational cost relative to classical solvers, and identify regimes where hybrid quantum models offer improved parameter efficiency. PINNACLE provides a foundation for benchmarking physics-informed learning methods and guiding future developments through quantitative assessment of their trade-offs.

Hyperbolic conservation laws govern a wide range of transport-driven dynamics featuring shocks, contact discontinuities, and complex wave interactions, posing distinct challenges for deep-learning-based surrogate modeling. While classical numerical methods provide robust and physically admissible solutions, their computational cost restricts applicability in many-query tasks such as parametric studies and design optimization. Conversely, existing neural surrogates offer rapid inference but often fail to respect intrinsic PDE structures, leading to non-physical artifacts, rollout instability, and poor generalization. We present an interpretable, structure-preserving graph neural solver that bridges classical numerical principles with graph neural networks (GNNs). The network is designed as a learned reconstruction-and-flux operator rather than a black-box state updater, thereby inherently preserving key properties such as local conservation and upwinding. Inspired by Arbitrary high-order DERivatives schemes, we further recast message-passing GNNs as high-order space-time predictors, enabling conservative and stable neural updates with large time steps. Evaluation is performed on challenging supersonic flow benchmarks spanning broad parametric variations in geometry, initial/boundary conditions, and flow regimes. The neural solver achieves superior long-horizon rollout stability and accuracy compared with strong surrogate baselines, outperforms low-order discretizations, and delivers orders-of-magnitude runtime speedups over high-resolution simulations.

Physics-aware recurrent convolutional networks (PARC) have demonstrated strong performance in predicting nonlinear spatiotemporal dynamics by embedding differential operators directly into the computational graph of a neural network. However, pixel-based convolutions are restricted to static, uniform Cartesian grids, making them ill-suited to following evolving localized structures in an efficient manner. Graph neural networks (GNNs) naturally handle irregular spatial discretizations, but existing graph-based physics-aware deep learning (PADL) methods have difficulty handling extreme nonlinear regimes. To address these limitations, we propose Graph PARC (G-PARC), which uses moving least squares (MLS) kernels to approximate spatial derivatives on unstructured graphs, and embeds the derivatives of governing partial differential equations into the network's computational graph. G-PARC achieves better accuracy with 2-3x fewer parameters than MeshGraphNet, MeshGraphKAN, and GraphSAGE, replacing the traditional encoder-processor-decoder framework with analytically computed differential operators. We demonstrate that G-PARC (1) generalizes across nonuniform spatial and temporal discretizations; (2) handles moving meshes required for structural deformation; and (3) outperforms existing graph-based PADL methods on nonlinear benchmarks including fluvial hydrology, planar shock waves, and elastoplastic dynamics. By embedding explicit physical operators within the flexibility of GNNs, G-PARC enables accurate modeling of extreme nonlinear phenomena on complex computational domains, moving PADLbeyond idealized Cartesian grids.

Accurate temperature estimation of pouch cells with indirect liquid cooling is essential for optimizing battery thermal management systems for transportation electrification. However, it is challenging due to the computational expense of finite element simulations and the limitations of data-driven models. This paper presents a physics-informed machine learning (PIML) framework for the efficient and reliable estimation of steady-state temperature profiles. The PIML approach integrates the governing heat transfer equations directly into the neural network's loss function, enabling high-fidelity predictions with significantly faster convergence than purely data-driven methods. The framework is evaluated on a dataset of varying cooling channel geometries. Results demonstrate that the PIML model converges more rapidly and achieves markedly higher accuracy, with a 49.1% reduction in mean squared error over the data-driven model. Validation against independent test cases further confirms its superior performance, particularly in regions away from the cooling channels. These findings underscore the potential of PIML for surrogate modeling and design optimization in battery systems.

Most practical engineering design problems involve nonlinear spatio-temporal dynamical systems. Multi-physics simulations are often performed to capture the fine spatio-temporal scales which govern the evolution of these systems. However, these simulations are often high-fidelity in nature, and can be computationally very expensive. Hence, generating data from these expensive simulations becomes a bottleneck in an end-to-end engineering design process. Spatio-temporal surrogate modeling of these dynamical systems has been a popular data-driven solution to tackle this computational bottleneck. This is because accurate machine learning models emulating the dynamical systems can be orders of magnitude faster than the actual simulations. However, one key limitation of purely data-driven approaches is their lack of generalizability to inputs outside the training distribution. In this paper, we propose a physics-informed spatio-temporal surrogate modeling (PISTM) framework constrained by the physics of the underlying dynamical system. The framework leverages state-of-the-art advancements in the field of Koopman autoencoders to learn the underlying spatio-temporal dynamics in a non-intrusive manner, coupled with a spatio-temporal surrogate model which predicts the behavior of the Koopman operator in a specified time window for unknown operating conditions. We evaluate our framework on a prototypical fluid flow problem of interest: two-dimensional incompressible flow around a cylinder.

Computational fluid dynamics (CFD) has become an essential tool for predicting fire behavior, yet maintaining both efficiency and accuracy remains challenging. A major source of computational cost in fire simulations is the modeling of radiation transfer, which is usually the dominant heat transfer mechanism in fires. Solving the high-dimensional radiative transfer equation (RTE) with traditional numerical methods can be a performance bottleneck. Here, we present a machine learning framework based on Fourier-enhanced multiple-input neural operators (Fourier-MIONet) as an efficient alternative to direct numerical integration of the RTE. We first investigate the performance of neural operator architectures for a small-scale 2D pool fire and find that Fourier-MIONet provides the most accurate radiative solution predictions. The approach is then extended to 3D CFD fire simulations, where the computational mesh is locally refined across multiple levels. In these high-resolution settings, monolithic surrogate models for direct field-to-field mapping become difficult to train and computationally inefficient. To address this issue, a nested Fourier-MIONet is proposed to predict radiation solutions across multiple mesh-refinement levels. We validate the approach on 3D McCaffrey pool fires simulated with FireFOAM, including fixed fire sizes and a unified model trained over a continuous range of heat release rates (HRRs). The proposed method achieves global relative errors of 2-4% for 3D varying-HRR scenarios while providing faster inference than the estimated cost of one finite-volume radiation solve in FireFOAM for the 16-solid-angle case. With fast and accurate inference, the surrogate makes higher-fidelity radiation treatments practical and enables the incorporation of more spectrally resolved radiation models into CFD fire simulations for engineering applications.

Surrogate modeling of body-driven fluid flows where immersed moving boundaries couple structural dynamics to chaotic, unsteady fluid phenomena remains a fundamental challenge for both computational physics and machine learning. We present AeTHERON, a heterogeneous graph neural operator whose architecture directly mirrors the structure of the sharp-interface immersed boundary method (IBM): a dual-graph representation separating fluid and structural domains, coupled through sparse cross-attention that reflects the compact support of IBM interpolation stencils. This physics-informed inductive bias enables AeTHERON to learn nonlinear fluid-structure coupling in a shared high-dimensional latent space, with continuous sinusoidal time embeddings providing temporal generalization across lead times. We evaluate AeTHERON on direct numerical simulations of a flapping flexible caudal fin, a canonical FSI benchmark featuring leading-edge vortex formation, large membrane deformation, and chaotic wake shedding across a 4x5 parameter grid of membrane thickness (h* = 0.01-0.04) and Strouhal number (St = 0.30-0.50). As a proof-of-concept, we train on the first 150 timesteps of a representative case using a 70/30 train/validation split and evaluate on the fully unseen extrapolation window t=150-200. AeTHERON captures large-scale vortex topology and wake structure with qualitative fidelity, achieving a mean extrapolation MAE of 0.168 without retraining, with error peaking near flapping half-cycle transitions where flow reorganization is most rapid -- a physically interpretable pattern consistent with the nonlinear fluid-membrane coupling. Inference requires milliseconds per timestep on a single GPU versus hours for equivalent DNS computation. This is a continuously developing preprint; results and figures will be updated in subsequent versions.

High-fidelity numerical simulation of subsurface flow is computationally intensive, especially for many-query tasks such as uncertainty quantification and data assimilation. Deep learning (DL) surrogates can significantly accelerate forward simulations, yet constructing them requires substantial machine learning (ML) expertise - from architecture design to hyperparameter tuning - that most domain scientists do not possess. Furthermore, the process is predominantly manual and relies heavily on heuristic choices. This expertise gap remains a key barrier to the broader adoption of DL surrogate techniques. For this reason, we present AutoSurrogate, a large-language-model-driven multi-agent framework that enables practitioners without ML expertise to build high-quality surrogates for subsurface flow problems through natural-language instructions. Given simulation data and optional preferences, four specialized agents collaboratively execute data profiling, architecture selection from a model zoo, Bayesian hyperparameter optimization, model training, and quality assessment against user-specified thresholds. The system also handles common failure modes autonomously, including restarting training with adjusted configurations when numerical instabilities occur and switching to alternative architectures when predictive accuracy falls short of targets. In our setting, a single natural-language sentence can be sufficient to produce a deployment-ready surrogate model, with minimum human intervention required at any intermediate stage. We demonstrate the utility of AutoSurrogate on a 3D geological carbon storage modeling task, mapping permeability fields to pressure and CO$_2$ saturation fields over 31 timesteps. Without any manual tuning, AutoSurrogate is able to outperform expert-designed baselines and domain-agnostic AutoML methods, demonstrating strong potential for practical deployment.

Neural operators have emerged as powerful surrogates for partial differential equation (PDE) solvers, yet they are typically trained as monolithic models for individual PDEs, require energy-intensive GPU hardware, and must be retrained from scratch when new physics emerge. We introduce the Spiking Compositional Neural Operator (SCNO), a modular architecture combining spiking and conventional components that addresses all three limitations. SCNO maintains a library of small spiking neural operator blocks, each trained on a single elementary differential operator (convection, diffusion, reaction), and composes them through a lightweight input-conditioned aggregator to solve coupled PDEs not seen during block training. A small correction network learns cross-coupling residuals while keeping all blocks and the aggregator frozen, preserving zero-forgetting modular expansion by construction. We evaluate SCNO on eight PDE families including five coupled systems and a nuclear-relevant 1-group neutron diffusion equation. SCNO with correction achieves the lowest relative $L^2$ error on four of five coupled PDEs, outperforming both a monolithic spiking DeepONet (by up to 62%, mean over 3 seeds) and a standard ANN DeepONet (by up to 65%), while requiring only 95K trainable parameters versus 462K for the monolithic baseline. To our knowledge, this is the first compositional spiking neural operator and the first proof-of-concept for modular neuromorphic PDE solving with built-in forgetting-free expansion.

We propose a hybrid physics-informed framework for solving families of parametric linear partial differential equations (PDEs) by combining a meta-learned predictor with a least-squares corrector. The predictor, termed KAPI (Kernel-Adaptive Physics-Informed meta-learner), is a shallow task-conditioned model that maps query coordinates and PDE parameters to solution values while internally generating an interpretable, task-adaptive Gaussian basis geometry. A lightweight meta-network maps PDE parameters to basis centers, widths, and activity patterns, thereby learning how the approximation space should adapt across the parametric family. This predictor-generated geometry is transferred to a second-stage corrector, which augments it with a background basis and computes the final solution through a one-shot physics-informed Extreme Learning Machine (PIELM)-style least-squares solve. We evaluate the method on four linear PDE families spanning diffusion, transport, mixed advection--diffusion, and variable-speed transport. Across these cases, the predictor captures meaningful physics through localized and transport-aligned basis placement, while the corrector further improves accuracy, often by one or more orders of magnitude. Comparisons with parametric PINNs, physics-informed DeepONet, and uniform-grid PIELM correctors highlight the value of predictor-guided basis adaptation as an interpretable and efficient strategy for parametric PDE solving.

Conditional Diffusion Models are powerful surrogates for emulating complex spatiotemporal dynamics, yet they often fail to match the accuracy of deterministic neural emulators for high-precision tasks. In this work, we address two critical limitations of autoregressive PDE diffusion models: their sub-optimal single-step accuracy and the prohibitive computational cost of unrolled training. First, we characterize the relationship between the noise schedule, the reconstruction error reduction rate and the diffusion exposure bias, demonstrating that standard schedules lead to suboptimal reconstruction error. Leveraging this insight, we propose an Adaptive Noise Schedule framework that minimizes inference reconstruction error by dynamically constraining the model's exposure bias. We further show that this optimized schedule enables a fast Proxy Unrolled Training method to stabilize long-term rollouts without the cost of full Markov Chain sampling. Both proposed methods enable significant improvements in short-term accuracy and long-term stability over diffusion and deterministic baselines on diverse benchmarks, including forced Navier-Stokes, Kuramoto-Sivashinsky and Transonic Flow.

Neural operators have emerged as powerful surrogates for dynamical systems due to their grid-invariant properties and computational efficiency. However, Fourier-based variants inherently truncate high-frequency components in spectral space, resulting in the loss of small-scale structures and degraded prediction quality at high resolutions when trained on low-resolution data. While diffusion-based enhancement methods can recover multi-scale features, they introduce substantial inference overhead that undermines the efficiency advantage of neural operators. In this work, we introduce MeanFlow-Enhanced Neural Operators (MENO), a novel framework that achieves accurate all-scale predictions with minimal inference cost. By leveraging the improved MeanFlow method, MENO restores both small-scale details and large-scale dynamics with superior physical fidelity and statistical accuracy. We evaluate MENO on three challenging dynamical systems, including phase-field dynamics, 2D Kolmogorov flow, and active matter dynamics, at resolutions up to 256$\times$256. Across all benchmarks, MENO improves the power spectrum density accuracy by up to a factor of 2 compared to baseline neural operators while achieving up to $14\times$ faster inference than the state-of-the-art Denoising Diffusion Implicit Model (DDIM)-enhanced counterparts, effectively bridging the gap between accuracy and efficiency. The flexibility and efficiency of MENO position it as an efficient surrogate model for scientific machine learning applications where both statistical integrity and computational efficiency are paramount.

Deep Learning Reduced Order Models (ROMs) are becoming increasingly popular as surrogate models for parametric partial differential equations (PDEs) due to their ability to handle high-dimensional data, approximate highly nonlinear mappings, and utilize GPUs. Existing approaches typically learn evolution either on the full solution field, which requires capturing long-range spatial interactions at high computational cost, or on compressed latent representations obtained from autoencoders, which reduces the cost but often yields latent vectors that are difficult to evolve, since they primarily encode spatial information. Moreover, in parametric PDEs, the initial condition alone is not sufficient to determine the trajectory, and most current approaches are not evaluated on jointly predicting multiple solution components with differing magnitudes and parameter sensitivities. To address these challenges, we propose a joint model consisting of a convolutional encoder, a transformer operating on latent representations, and a decoder for reconstruction. The main novelties are joint training with multi-stage parameter injection and coordinate channel injection. Parameters are injected at multiple stages to improve conditioning. Physical coordinates are encoded to provide spatial information. This allows the model to dynamically adapt its computations to the specific PDE parameters governing each system, rather than learning a single fixed response. Experiments on the Advection-Diffusion-Reaction equation and Navier-Stokes flow around the cylinder wake demonstrate that our approach combines the efficiency of latent evolution with the fidelity of full-field models, outperforming DL-ROMs, latent transformers, and plain ViTs in multi-field prediction, reducing the relative rollout error by approximately $5$ times.

The impact of wave-induced forcing on the mean water level and nearshore currents is typically modeled through excess momentum fluxes, also known as radiation stresses, and their spatial gradients. Accurate storm surge prediction requires coupled circulation and wave models, but the high computational cost of numerical wave models limits their temporal resolution. In this work, we explore a proof-of-concept application of Deep Operator Networks (DeepONets) as a surrogate for the Simulating WAves Nearshore (SWAN) numerical wave model. Unlike grid-dependent surrogate models, DeepONets learn the underlying continuous operator, and thus, can provide highly efficient prediction while enabling discretization-invariant inference. The proposed surrogate model is evaluated using two distinct 1-D and 2-D steady-state numerical examples with variable boundary wave conditions and wind fields. When applied to a realistic numerical example of steady-state wave simulation in Duck, NC, the DeepONet surrogate improves computational efficiency by four orders of magnitude. Furthermore, the model demonstrates consistently high accuracy in predicting the significant wave height and the x- and y- components of the radiation stress gradient, by achieving relative L_2 errors bounded by 1.91%, 10.98%, and 6.88%, respectively, across all unseen test scenarios.

Cardiovascular modeling has rapidly advanced over the past few decades due to the rising needs for health tracking and early detection of cardiovascular diseases. While 1-D arterial models offer an attractive compromise between computational efficiency and solution fidelity, their application on large populations or for generating large in silico cohorts remains challenging. Certain hemodynamic parameters like the terminal resistance/compliance, are difficult to clinically estimate and often yield non-physiological hemodynamics when sampled naively, resulting in large portions of simulated datasets to be discarded. In this work, we present a systematic framework for training machine learning (ML) models, capable of instantaneous hemodynamic prediction and parameter estimation. We initially start with generating a parametric virtual cohort of patients which is based on the multivariate correlations observed in the large Asklepios clinical dataset, ensuring that physiological parameter distributions are respected. We then train a deep neural surrogate model, able to predict patient-specific arterial pressure and cardiac output (CO), enabling rapid a~priori screening of input parameters. This allows for immediate rejection of non-physiological combinations and drastically reduces the cost of targeted synthetic dataset generation (e.g. hypertensive groups). The model also provides a principled means of sampling the terminal resistance to minimize the uncertainties of unmeasurable parameters. Moreover, by assessing the model's predictive performance we determine the theoretical information which suffices for solving the inverse problem of estimating the CO. Finally, we apply the surrogate on a clinical dataset for the estimation of central aortic hemodynamics i.e. the CO and aortic systolic blood pressure (cSBP).

Operational flood forecasting still relies on high-fidelity two-dimensional hydraulic solvers, but their runtime can be prohibitive for rapid decision support on large urban floodplains. In parallel, AI-based surrogate models have shown strong potential in several areas of computational physics for accelerating otherwise expensive high-fidelity simulations. We address this issue on the lower Têt River (France), starting from a production-grade Telemac2D model defined on a high-resolution unstructured finite-element mesh with more than $4\times 10^5$ nodes. From this setup, we build a learning-ready database of synthetic but operationally grounded flood events covering several representative hydrograph families and peak discharges. On top of this database, we develop a graph-neural surrogate based on projected meshes and multimesh connectivity. The projected-mesh strategy keeps training tractable while preserving high-fidelity supervision from the original Telemac simulations, and the multimesh construction enlarges the effective spatial receptive field without increasing network depth. We further study the effect of an explicit discharge feature $Q(t)$ and of pushforward training for long autoregressive rollouts. The experiments show that conditioning on $Q(t)$ is essential in this boundary-driven setting, that multimesh connectivity brings additional gains once the model is properly conditioned, and that pushforward further improves rollout stability. Among the tested configurations, the combination of $Q(t)$, multimesh connectivity, and pushforward provides the best overall results. These gains are observed both on hydraulic variables over the surrogate mesh and on inundation maps interpolated onto a common $25\,\mathrm{m}$ regular grid and compared against the original high-resolution Telemac solution. On the studied case, the learned surrogate produces 6-hour predictions in about $0.4\,\mathrm{s}$ on a single NVIDIA A100 GPU, compared with about $180\,\mathrm{min}$ on 56 CPU cores for the reference simulation. These results support graph-based surrogates as practical complements to industrial hydraulic solvers for operational flood mapping.

Severe accident analysis using system-level codes such as MELCOR is indispensable for nuclear safety assessment, yet the computational cost of repeated simulations poses a significant bottleneck for parametric studies and uncertainty quantification. Existing surrogate models accelerate these analyses but depend on large volumes of simulation data, while physics-informed neural networks (PINNs) enable data-free training but must be retrained for every change in problem parameters. This study addresses both limitations by developing the Parameterized PINNs coupled with FDM (P2F) method, a node-assigned hybrid framework for MELCOR's Control Volume Hydrodynamics/Flow Path (CVH/FP) module. In the P2F method, a parameterized Node-Assigned PINN (NA-PINN) accepts the water-level difference, initial velocity, and time as inputs, learning a solution manifold so that a single trained network serves as a data-free surrogate for the momentum conservation equation across all flow paths without retraining. This PINN is coupled with a finite difference method (FDM) solver that advances the mass conservation equation at each time step, ensuring exact discrete mass conservation while replacing the iterative nonlinear momentum solve with a single forward pass. Verification on a six-tank gravity-driven draining scenario yields a water level mean absolute error of $7.85 \times 10^{-5}$ m and a velocity mean absolute error of $3.21 \times 10^{-3}$ m/s under the nominal condition with $Δt = 1.0$ s. The framework maintains consistent accuracy across time steps ranging from 0.2 to 1.0 s and generalizes to five distinct initial conditions, all without retraining or simulation data. This work introduces a numerical coupling methodology for integrating parameterized PINNs with FDM within a nuclear thermal-hydraulic system code framework.

Computational fluid dynamics (CFD) simulations of complex fluid flows in energy systems are prohibitively expensive due to strong nonlinearities and multiscale-multiphysics interactions. In this work, we present a transformer-based modeling framework for prediction of fluid flows, and demonstrate it for high-pressure gas injection phenomena relevant to reciprocating engines. The approach employs a hierarchical Vision Transformer (SwinV2-UNet) architecture that processes multimodal flow datasets from multi-fidelity simulations. The model architecture is conditioned on auxiliary tokens explicitly encoding the data modality and time increment. Model performance is assessed on two different tasks: (1) spatiotemporal rollouts, where the model autoregressively predicts the flow state at future times; and (2) feature transformation, where the model infers unobserved fields/views from observed fields/views. We train separate models on multimodal datasets generated from in-house CFD simulations of argon jet injection into a nitrogen environment, encompassing multiple grid resolutions, turbulence models, and equations of state. The resulting data-driven models learn to generalize across resolutions and modalities, accurately forecasting the flow evolution and reconstructing missing flow-field information from limited views. This work demonstrates how large vision transformer-based models can be adapted to advance predictive modeling of complex fluid flow systems.

Reservoir simulation workflows face a fundamental data asymmetry: input parameter fields (geostatistical permeability realizations, porosity distributions) are free to generate in arbitrary quantities, yet existing neural operator surrogates require large corpora of expensive labeled simulation trajectories and cannot exploit this unlabeled structure. We introduce PI-JEPA (Physics-Informed Joint Embedding Predictive Architecture), a surrogate pretraining framework that trains without any completed PDE solves, using masked latent prediction on unlabeled parameter fields under per-sub-operator PDE residual regularization. The predictor bank is structurally aligned with the Lie--Trotter operator-splitting decomposition of the governing equations, dedicating a separate physics-constrained latent module to each sub-process (pressure, saturation transport, reaction), enabling fine-tuning with as few as 100 labeled simulation runs. On single-phase Darcy flow, PI-JEPA achieves $1.9\times$ lower error than FNO and $2.4\times$ lower error than DeepONet at $N_\ell{=}100$, with 24% improvement over supervised-only training at $N_\ell{=}500$, demonstrating that label-free surrogate pretraining substantially reduces the simulation budget required for multiphysics surrogate deployment.

We consider the problem of constructing surrogate operators for parameter-to-solution maps arising from parametric partial differential equations, where repeated forward model evaluations are computationally expensive. We present a systematic empirical comparison of neural operator surrogates, including a reduced-basis neural operator trained with $L^2_μ$ and $H^1_μ$ objectives and the Fourier neural operator, against polynomial surrogate methods, specifically a reduced-basis sparse-grid surrogate and a reduced-basis tensor-train surrogate. All methods are evaluated on a linear parametric diffusion problem and a nonlinear parametric hyperelasticity problem, using input fields with algebraically decaying spectral coefficients at varying rates of decay $s$. To enable fair comparisons, we analyze ensembles of surrogate models generated by varying hyperparameters and compare the resulting Pareto frontiers of cost versus approximation accuracy, decomposing cost into contributions from data generation, setup, and evaluation. Our results show that no single method is universally superior. Polynomial surrogates achieve substantially better data efficiency for smooth input fields ($s \geq 2$), with convergence rates for the sparse-grid surrogate in agreement with theoretical predictions. For rough inputs ($s \leq 1$), the Fourier neural operator displays the fastest convergence rates. Derivative-informed training consistently improves data efficiency over standard $L^2_μ$ training, providing a competitive alternative for rough inputs in the low-data regime when Jacobian information is available at reasonable cost. These findings highlight the importance of matching the surrogate methodology to the regularity of the problem as well as accuracy demands and computational constraints of the application.

Production optimization in stress-sensitive unconventional reservoirs is governed by a nonlinear trade-off between pressure-driven flow and stress-induced degradation of fracture conductivity and matrix permeability. While higher drawdown improves short-term production, it accelerates permeability loss and reduces long-term recovery. Identifying optimal, time-varying control strategies requires repeated evaluations of fully coupled flow-geomechanics simulators, making conventional optimization computationally expensive. We propose a deep learning-based surrogate optimization framework for high-dimensional well control. Unlike prior approaches that rely on predefined control parameterizations or generic sampling, our method treats well control as a continuous, high-dimensional problem and introduces a problem-informed sampling strategy that aligns training data with trajectories encountered during optimization. A neural network proxy is trained to approximate the mapping between bottomhole pressure trajectories and cumulative production using data from a coupled flow-geomechanics model. The proxy is embedded within a constrained optimization workflow, enabling rapid evaluation of control strategies. Across multiple initializations, the surrogate achieves agreement with full-physics solutions within 2-5 percent, while reducing computational cost by up to three orders of magnitude. Discrepancies are mainly associated with trajectories near the boundary of the training distribution and local optimization effects. This framework shows that combining surrogate modeling with problem-informed sampling enables scalable and reliable optimization for high-dimensional, simulator-based problems, with broader applicability to PDE-constrained systems.

We develop a machine learning (ML) surrogate model to approximate solutions to Maxwell's equations in one dimension, focusing on scenarios involving a material interface that reflects and transmits electro-magnetic waves. Derived from high-fidelity Finite Volume (FV) simulations, our training data includes variations of the initial conditions, as well as variations in one material's speed of light, allowing for the model to learn a range of wave-material interaction behaviors. The ML model autoregressively learns both the physical and frequency embeddings in a vision transformer-based framework. By incorporating Fourier transforms in the latent space, the wave number spectra of the solutions aligns closely with the simulation data. Prediction errors exhibit an approximately linear growth over time with a sharp increase at the material interface. Test results show that the ML solution has adequate relative errors below $10\%$ in over $75$ time step rollouts, despite the presence of the discontinuity and unknown material properties.

For neural surrogates of time-dependent partial differential equations (PDEs), storage and memory budgets require training data to be obtained by coarsening and quantizing very-high-resolution numerical simulations. Fine-scale information can thus be lost before it reaches the model. We analyze this effect in periodic incompressible Navier-Stokes: quantization spreads error evenly over the retained modes, so the finest retained band, which holds the least energy, is the most corrupted, and velocity and vorticity are distorted differently under the same operator. Building on this analysis, we propose Derived-Field Optimization (DerivOpt), which chooses which primitive and derived fields to carry from a supplied library, and how many bits each receives, by exactly minimizing a closed-form posterior risk under a calibrated channel model. This optimization problem always has an optimum and requires no knowledge of the governing equation. On PDEBench, DerivOpt not only lowers the pooled rollout nRMSE but also substantially reduces the fine-band error. These gains are already present in the predictor-free input state and add to those of a multiscale architecture. Which fields are carried, and at what precision, is thus an optimizable design variable upstream of the predictor that determines how much information a fixed budget leaves recoverable.

The non-equilibrium dynamics of mesoscale phase transitions are fundamentally shaped by thermal fluctuations, which not only seed instabilities but actively control kinetic pathways, including rare barrier-crossing events such as nucleation that are entirely inaccessible to deterministic models. Machine-learning surrogates for such systems must therefore represent stochasticity explicitly, enforce conservation laws by construction, and expose physically interpretable structure. We develop physics-aware surrogate models for the stochastic Cahn-Hilliard equation in 3D that satisfy all three requirements simultaneously. The key innovation is to parameterize the surrogate at the level of inter-cell fluxes, decomposing each flux into a deterministic mobility-weighted chemical-potential gradient and a learnable noise amplitude. This design guarantees exact mass conservation at every step and adds physical fluctuations to inter-cell mass transport. A learnable free energy functional provides thermodynamic interpretability, validated by independent recovery of the bulk double-well landscape, interfacial excess energy, and curvature-independent interfacial tension. Tests demonstrate accurate reproduction of ensemble statistics and noise-accelerated coarsening, with generalization to spatial domains 64 times larger in volume and temporal horizons 160x longer than those seen during training. Critically, the stochastic surrogate captures thermally activated nucleation in the metastable regime, a qualitative capability that no deterministic surrogate can provide regardless of training, thus establishing flux-level stochasticity as an architectural necessity rather than an optional enhancement.

Computational fluid dynamics (CFD) provides high-fidelity simulations of fluid flows but remains computationally expensive for many-query applications. In recent years deep learning (DL) has been used to construct data-driven fluid-dynamic surrogate models. In this work we consider a different learning paradigm and embrace generative modelling as a framework for constructing scalable fluid-dynamics surrogate models. We introduce FluidFlow, a generative model based on conditional flow-matching, a recent alternative to diffusion models that learns deterministic transport maps between noise and data distributions. FluidFlow is specifically designed to operate directly on CFD data defined on both structured and unstructured meshes alike, without the needs to perform any mesh interpolation pre-processing and preserving geometric fidelity. We assess the capabilities of FluidFlow using two different core neural network architectures, a U-Net and diffusion transformer (DiT), and condition their learning on physically meaningful parameters. The methodology is validated on two benchmark problems of increasing complexity: prediction of pressure coefficients along an airfoil boundary across different operating conditions, and prediction of pressure and friction coefficients over a full three-dimensional aircraft geometry discretized on a large unstructured mesh. In both cases, FluidFlow outperform strong multilayer perceptron baselines, achieving significantly lower error metrics and improved generalisation across operating conditions. Notably, the transformer-based architecture enables scalable learning on large unstructured datasets while maintaining high predictive accuracy. These results demonstrate that flow-matching generative models provide an effective and flexible framework for surrogate modelling in fluid dynamics, with potential for realistic engineering and scientific applications.

Training reinforcement learning (RL) agents to control fluid dynamics systems is computationally expensive due to the high cost of direct numerical simulations (DNS) of the governing equations. Surrogate models offer a promising alternative by approximating the dynamics at a fraction of the computational cost, but their feasibility as training environments for RL is limited by distribution shifts, as policies induce state distributions not covered by the surrogate training data. In this work, we investigate the use of Linear Recurrent Autoencoder Networks (LRANs) for accelerating RL-based control of 2D Rayleigh-Bénard convection. We evaluate two training strategies: a surrogate trained on precomputed data generated with random actions, and a policy-aware surrogate trained iteratively using data collected from an evolving policy. Our results show that while surrogate-only training leads to reduced control performance, combining surrogates with DNS in a pretraining scheme recovers state-of-the-art performance while reducing training time by more than 40%. We demonstrate that policy-aware training mitigates the effects of distribution shift, enabling more accurate predictions in policy-relevant regions of the state space.

Graph-based surrogate models provide fast alternatives to high-fidelity CFD solvers, but their opaque latent spaces and limited controllability restrict use in safety-critical settings. A key failure mode in oscillatory flows is phase drift, where predictions remain qualitatively correct but gradually lose temporal alignment with observations, limiting use in digital twins and closed-loop control. Correcting this through retraining is expensive and impractical during deployment. We ask whether phase drift can instead be corrected post hoc by manipulating the latent space of a frozen surrogate. We propose a phase-steering framework for pretrained graph-based CFD models that combines the right representation with the right intervention mechanism. To obtain disentangled representation for effective steering, we use sparse autoencoders (SAEs) on frozen MeshGraphNet embeddings. To steer dynamics, we move beyond static per-feature interventions such as scaling or clamping, and introduce a temporally coherent, phase-aware method. Specifically, we identify oscillatory feature pairs with Hilbert analysis, project spatial fields into low-rank temporal coefficients via SVD, and apply smooth time-varying rotations to advance or delay periodic modes while preserving amplitude-phase structure. Using a representation-agnostic setup, we compare SAE-based steering with PCA and raw embedding spaces under the same intervention pipeline. Results show that sparse, disentangled representations outperform dense or entangled ones, while static interventions fail in this dynamical setting. Overall, this work shows that latent-space steering can be extended from semantic domains to time-dependent physical systems when interventions respect the underlying dynamics, and that the same sparse features used for interpretability can also serve as physically meaningful control axes.

Air flow modeling at a local scale is essential for applications such as pollutant dispersion modeling or wind farm modeling. To circumvent costly Computational Fluid Dynamics (CFD) computations, deep learning surrogate models have recently emerged as promising alternatives. However, in the context of urban air flow, deep learning models struggle to adapt to the high variations of the urban geometry and to large mesh sizes. To tackle these challenges, we introduce Anchored Branched Steady-state WInd Flow Transformer (AB-SWIFT), a transformer-based model with an internal branched structure uniquely designed for atmospheric flow modeling. We train our model on a specially designed database of atmospheric simulations around randomised urban geometries and with a mixture of unstable, neutral, and stable atmospheric stratifications. Our model reaches the best accuracy on all predicted fields compared to state-of-the-art transformers and graph-based models. Our code and data is available at https://github.com/cerea-daml/abswift.