arXiv subjects

George Em Karniadakis

Publications and source records attributed to George Em Karniadakis.

At least 19 recordsLinked to original sources

arXiv2026

Well-posedness of neural turbulence closures and tangent dissipation

A neural turbulence closure defines a new boundary-value problem, $R(U)=N(U)+F(U)=0$, with a coupled Jacobian $J(U)=N'(U)+F'(U)$, where $N$ is the original mean-flow operator and $F$ the learned closure. We establish two consequences of global tangent dissipation. For a monotone original operator, a positive uniform margin supplied by the original operator and closure together guarantees existence, uniqueness and a global inverse-sensitivity bound relating a posteriori solution error to the a priori residual. For a general original operator, a dissipative closure cannot worsen tangent dissipation, but this alone does not guarantee uniqueness. Tangent dissipation depends on both diffusion and reaction. We study two complementary ways to promote it: (1) an exact-integral construction enforcing non-negative tangent diffusion while leaving reaction unconstrained, and (2) a penalty on tangent-reaction violations at sampled states. Tangent diffusion enters the Jacobian, and non-negative secant eddy viscosity alone does not control its coercivity. We conduct tests with channel flow at $Re_τ=180$--$5200$, which provides a strongly monotone baseline. Both constrained closures reach accurate solutions in all 50 training-seed/Reynolds-number cases. At $Re_τ=1000$, we conduct tests with 10,000 starts for one fixed network per closure and we find one root for each constrained closure and multiple roots for the other closures. Although this does not prove uniqueness, it provides strong empirical evidence for uniqueness of the tested constrained closures. At $Re_τ=5200$, the construction and penalty reduce the reported inverse sensitivity relative to the original operator by approximately $372\times$ and $11\times$, respectively.

physics.flu-dyn↗

arXiv2026

GRAFT-ATHENA: Self-Improving Agentic Teams for Autonomous Discovery and Evolutionary Numerical Algorithms

Scientific methods are developed for classes of problems, so knowledge transfers across structurally related cases. Language-model agents can execute scientific workflows, but their problem--method relationships remain implicit, so each new problem restarts the search and little of what worked transfers. We introduce GRAFT--ATHENA, which makes this problem-to-method map explicit as an expandable probabilistic structure of admissible problems, methods, and their dependencies. Graph factorization keeps the substrate tractable, and semantic fingerprints measure similarity, so experience guides related problems. As a result, the framework matched or exceeded expert baselines, attaining near-machine-precision losses in physics-informed learning, reproducing clinically consistent blood-rheology trends, and developing a high-order hypersonic-flow solver for the Apollo Command Module that matched experimental measurements within $1.8\%$. It also proposed a certified regularization for ill-posed in vivo brain-flow reconstruction, developed a spectrally convergent physics-informed architecture, and established machine-checked universal-approximation theorems for two widely used architectures. Scientific structure enables cumulative and verifiable agentic discovery.

cs.LG↗

arXiv2026

Quantifying the Biophysical Properties of Red Blood Cells in Gaucher Disease

Gaucher disease (GD), the most common lysosomal storage disorder, alters red blood cell (RBC) mechanics and circulation, contributing to vascular occlusions, bone infarcts, and splenomegaly. However, the individual roles of GD-RBC biophysical properties in these processes remain unclear. Here, we present a combined computational-experimental investigation to quantitatively characterize GD-RBC biophysical properties and determine how specific mechanical parameters drive abnormal RBC behavior. Informed by experimental data, we independently quantify key RBC properties, including shear modulus (mu), surface-to-volume ratio (S/V), and bending modulus (k_c). Based on these parameters, we construct three GD-RBC subtypes (GD-RBC1-3) to systematically isolate their individual contributions. At the single-cell level, optical tweezers simulations show up to ~27% reduction in axial diameter and ~42% reduction in transverse compression. Tank-treading dynamics exhibit non-monotonic behavior, with rotation frequencies increasing by up to ~70% or decreasing under elevated bending rigidity. In confined flow, traversal times through microchannel constrictions increase by more than a factor of two, while splenic slit passage times rise from ~250 ms (control) to >1200 ms for the severe GD-RBC subtype, approaching a functional no-passage threshold. At the population level, viscosity simulations demonstrate that these alterations collectively elevate blood viscosity, with small fractions (~4.0%) of highly rigid cells disproportionately increasing flow resistance. Overall, this study provides a quantitative and mechanistic framework that disentangles the contributions of key RBC parameters to abnormal behavior in GD, linking cellular-scale biophysics to hematologic dysfunction and microvascular occlusion.

physics.bio-ph↗

arXiv2026

Residual-based attention in physics-informed neural networks

Driven by the need for more efficient and seamless integration of physical models and data, physics-informed neural networks (PINNs) have seen a surge of interest in recent years. However, ensuring the reliability of their convergence and accuracy remains a challenge. In this work, we propose an efficient, gradient-less weighting scheme for PINNs that accelerates the convergence of dynamic or static systems. This simple yet effective attention mechanism is a bounded function of the evolving cumulative residuals and aims to make the optimizer aware of problematic regions at no extra computational cost or adversarial learning. We illustrate that this general method consistently achieves one order of magnitude faster convergence than vanilla PINNs and a minimum relative $L^{2}$ error of $\mathcal{O}(10^{-5})$, on typical benchmarks of the literature. The method is further tested on the inverse solution of the Navier-Stokes within the brain perivascular spaces, where it considerably improves the prediction accuracy. Furthermore, an ablation study is performed for each case to identify the contribution of the components that enhance the vanilla PINN formulation. Evident from the convergence trajectories is the ability of the optimizer to effectively escape from poor local minima or saddle points while focusing on the challenging domain regions, which consistently have a high residual score. We believe that alongside exact boundary conditions and other model reparameterizations, this type of attention mask could be an essential element for fast training of both PINNs and neural operators.

cs.LG↗

arXiv2026

Learning in PINNs: Phase transition, diffusion equilibrium, and generalization

We investigate the learning dynamics of fully-connected neural networks through the lens of the neural gradient signal-to-noise ratio (SNR), examining the behavior of first-order optimizers in non-convex objectives. Interpreting the drift/diffusion phases as proposed in the information bottleneck theory, we identify a third phase termed "diffusion equilibrium" (DE), a stable training phase characterized by highly-ordered neural gradients across the sample space. This phase is marked by an abrupt transition, where sample-wise gradients align (SNR increases), and stable optimizer convergence. Moreover, we find that when homogeneous residuals are also met across the sample space during the DE phase, this leads to better generalization, as the optimization steps are equally sensitive to each sample. Based on this observation, we propose a sample-wise re-weighting scheme, which considerably improves the residual homogeneity and generalization in quadratic loss functions, by targeting the problematic samples with large residuals and vanishing gradients. Finally, we explore the information compression phenomenon, pinpointing a significant saturation-induced compression of activations at the DE phase transition, driven by the sample-wise gradient directional alignment. Interestingly, it is during the saturation of activations that the model converges, with deeper layers experiencing negligible information loss. Supported by experimental examples on physics-informed neural networks (PINNs), which highlight the critical role of gradient agreement due to their inherent PDE-based interdependence of samples, our findings suggest that when both sample-wise gradients and residuals are ordered, this leads to faster convergence and better generalization. Identifying phase transitions could improve deep learning optimization strategies, enhancing physics-informed methods and machine learning performance.

cs.LG↗

arXiv2026

Adjoint Method versus Physics-Informed Neural Networks in PDE-Constrained Inverse Problems

Inverse problems governed by partial differential equations (PDEs) are central to computational mechanics and are commonly solved by adjoint-based optimization, while physics-informed neural networks (PINNs) have emerged as a flexible alternative. Their relative performance remains difficult to assess because the two approaches are often compared under different formulations, parameterizations, optimizers, and regularization choices. We present a controlled comparison of adjoint optimization and PINNs for PDE-constrained inverse problems. From a common abstract formulation, we instantiate both methods on identical domains, governing equations, observation models, and regularization terms, while matching the optimizer, unknown parameterization, and arithmetic precision wherever applicable. The benchmarks include unsteady Burgers, noisy Darcy permeability inversion, three-dimensional Allen-Cahn reaction identification, and unsteady Navier-Stokes viscosity identification. The results show that the representation of the unknown largely determines the preferred method: grid-based fields favor the discrete adjoint, whereas neural representations are native to PINNs and relevant for closure and constitutive modeling. For time-dependent problems, adjoint inversion can be dominated by trajectory storage and differentiation, while PINNs provide satisfactory reconstructions at lower cost. A PINN-warm-started adjoint strategy then recovers adjoint-level accuracy at about half the cost.

math.NA↗

arXiv2026

Spectrally Safe Neural Operator Warm-Starts for Large-Scale Newton Solvers

Neural operators are increasingly used to warm-start Newton solvers for nonlinear PDEs, on the premise that a low test error places the initial guess inside the basin of attraction. We show that this premise is unreliable. An operator trained to the relative $L^2$ error $O(10^{-3})$ can still produce an initial state in which the discrete Jacobian is indefinite, because the mean-squared training controls error on average while leaving localized pointwise violations of the underlying physics. For a nearly incompressible hyperelasticity problem, we trace this to the predicted volume change: the operator disperses $\mathrm{det} F$ well away from one, and the resulting Jacobian acquires negative eigenvalues even when the predicted field is visually indistinguishable from the reference. At a small scale, this is a nuisance; at a multi-million degree-of-freedom scale, it is disqualifying, since the conjugate gradient and other Krylov solvers needed for memory-feasible Newton steps assume a definite spectrum. We then show that a short, label-free fine-tuning phase -- penalizing the operator against the discrete energy, with no additional solution data -- shifts the Jacobian spectrum back to positive definite. Combined with an inexact outer loop, this gives a warm-started Newton method that converges across the full loading range where the unregularized operator fails, reaching up to 5.4$\times$ wall-clock speedup over incremental continuation on a 3D problem with 6.4 million degrees of freedom.

math.NA↗

arXiv2026

Physics-Informed Laplace Neural Operator for Solving Partial Differential Equations

Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs). However, purely data-driven models often require extensive training data and can generalize poorly, especially in small-data regimes and under unseen (out-of-distribution) input functions that are not represented in the training data. To address these limitations, we propose the Physics-Informed Laplace Neural Operator (PILNO), which enhances the Laplace Neural Operator (LNO) by embedding governing physics into training through PDE, boundary condition, and initial condition residuals. To improve expressivity, we first introduce an Advanced LNO (ALNO) backbone that retains a pole-residue transient representation while replacing the steady-state branch with an FNO-style Fourier multiplier. To make physics-informed training both data-efficient and robust, PILNO further leverages (i) virtual inputs: an unlabeled ensemble of input functions spanning a broad spectral range that provides abundant physics-only supervision and explicitly targets out-of-distribution (OOD) regimes; and (ii) temporal-causality weighting: a time-decaying reweighting of the physics residual that prioritizes early-time dynamics and stabilizes optimization for time-dependent PDEs. Across four representative benchmarks -- Burgers' equation, Darcy flow, a reaction-diffusion system, and a forced KdV equation -- PILNO consistently improves accuracy in small-data settings (e.g., N_train <= 27), reduces run-to-run variability across random seeds, and achieves stronger OOD generalization with respect to input function statistics than purely data-driven baselines.

cs.LG↗

arXiv2026

Fixed and Adaptive Topological DeepONets: Functional Measurements on Hausdorff Locally Convex Spaces

Deep Operator Networks (DeepONets; arXiv:1910.03193) typically encode an input function through point values on a fixed discretization. Building on the Topological DeepONet framework of Ismailov (arXiv:2603.11972), we replace point samples by continuous linear functionals drawn from the continuous dual of a Hausdorff locally convex space $({V},\{p_α\}_{α\in A})$, whose topology is generated by a point-separating family of seminorms rather than a single norm, and develop fixed and adaptive functional measurement systems. Measurements are combined with the coefficient-space Two-Step procedure of Lee and Shin (arXiv:2309.01020), while a training-only decoder and regularization stabilize the adaptive coordinates. We derive a discrete error decomposition separating measurement, output-basis, and neural-approximation errors, together with a Barron-rate refinement. The framework is evaluated on the antiderivative operator, a non-normable locally convex input space, heterogeneous Darcy flow, a controlled operator, and fixed-time and time-evolving Navier-Stokes vorticity operators. In the heterogeneous Darcy problem, the functional models retain nearly resolution-independent errors of 5.5-5.6% on unseen grids, while in the controlled problem adaptive measurements reduce the mean error below 1.2%. For the fixed-time Navier-Stokes problem, the Adaptive Topological DeepONet is the most accurate DeepONet-based model, attaining a mean relative $L^2$ error of 1.685% +/- 0.017% using 128 functional coordinates. A comparably sized Fourier neural operator (FNO; arXiv:2010.08895) achieves the lower error 0.832% +/- 0.172%, but requires the full 64x64 input field, twice the training time, and 10.7x greater peak GPU memory. The formulation provides compact, interpretable, and discretization-portable coordinates in the continuous dual $V'$, including for non-normable input spaces.

cs.LG↗

arXiv2026

Quantifying reticulocyte biomechanics in health and disease

Red blood cell (RBC) populations are mechanically heterogeneous, yet how this shapes transport, clogging, and rheology in confined environments remains unclear. We combine microfluidic microchannel experiments with dissipative particle dynamics (DPD) simulations to study how reticulocyte morphology, deformability, and cell-cell hydrodynamic coupling govern microconfined blood flow, and link these to acute and chronic mountain sickness. Reticulocyte-rich samples show three subtypes (multilobular, cup-shaped, near-discocytic), parameterized (R1-R3) by fitting microchannel transit and shape-under-flow data. Single-cell simulations show that 5-micron microchannels amplify mechanical heterogeneity (R1 transits 30-50% more slowly than softer cells), whereas bending-dominated splenic slits discriminate subtypes by only 10-20%. Pairwise simulations show that a leading cell never lets a follower pass below its own single-cell threshold - so the order-of-magnitude, wake-"unjamming" reduction is absent - but the leader's compliance shapes crowded single-file passage: a soft reticulocyte leader lowers a trailing stiff cell's critical passage pressure by ~12% relative to a stiff (sickle-trait) leader and speeds its transit by ~10%. The controlling variable is the single-cell critical pressure gradient Delta_P_c, which rises monotonically with membrane stiffness from control discocytes through reticulocytes to sickle-cell-trait cells. Our simulations reproduce the shear-thinning viscosity of control blood, against which the reported chronic-mountain-sickness hyperviscosity reflects predominantly hematocrit-driven crowding rather than a change in single-cell rheology. These results place benign acclimatization, chronic-mountain-sickness hyperviscosity, and sickle-cell-trait splenic syndrome on a single mechanical axis defined by Delta_P_c relative to the splenic operating pressure.

cond-mat.soft↗

arXiv2026

Generalizable turbulence closures across bluff-body shapes by PINN-based solver-agnostic training

Data-driven turbulence closures are usually calibrated by inverse methods that embed a CFD solver in the loop, tying the model to a particular discretization and requiring every iterate to yield a convergent solve. We instead train the closure inside a physics-informed neural network (PINN): the Reynolds-averaged Navier-Stokes residual is imposed by automatic differentiation, so the inverse problem is mesh-free, differentiable, and solver-agnostic. Because no forward solve runs during training, only the final closure need be solver-stable, arbitrary neural closures are admitted without an adjoint, and the iterative cost of adjoint or ensemble methods vanishes; each hypothesis trains in minutes on a single GPU, so the framework rapidly screens closure forms. We develop four closures: three model the Reynolds stress on a realizable tensor basis -- a local map, a non-local model transporting the turbulent kinetic energy and recovering the out-of-plane normal stress, and the same with a learned length scale l -- and a fourth models the Reynolds force F = -\nabla \cdot τdirectly, free of the realizability constraint. All four are trained across six two-dimensional bluff-body wakes at Re = 10^4 and deployed frozen in a standard finite-element solver, stabilized by input-gradient smoothing and a Lipschitz constraint. Under a strict leave-one-shape-out (LOSO) protocol, all four improve substantially on a steady SST k-omega baseline. The learned-length-scale closure is most accurate on the stress fields, while the force model generalizes best on the mean velocity and drag (LOSO drag error ~8.5%). The closures also train efficiently on Particle Image Velocimetry data, enabling geometries intractable for DNS.

physics.flu-dyn↗

arXiv2026

ATHENA: Agentic Team for Hierarchical Evolutionary Numerical Algorithms

Progress in computational science depends on complex numerical workflows that must faithfully encode physical laws, yet translating conceptual insight into reliable code remains a major bottleneck. Although large language models can generate isolated code fragments, they lack the structured reasoning required to design, verify, and iteratively refine complete scientific pipelines. Here we introduce ATHENA, an agentic framework explicitly designed to emulate scientific research modeled as a knowledge-driven contextual bandit process. Its core loop separates conceptual policy from numerical realization through expert-derived conceptual scaffolding, enabling principled diagnosis, reformulation, and repair of computational strategies. Across scientific computing and scientific machine learning tasks, ATHENA autonomously derives and correctly applies exact analytical solutions, constructs stable numerical solvers, diagnoses ill-posed formulations, and orchestrates hybrid symbolic-numeric workflows. Quantitatively, ATHENA matches and frequently surpasses the accuracy of expert-authored reference solutions reported in the literature on canonical benchmarks. By reframing computation as an object of agentic reasoning, our framework enables autonomous orchestration of heterogeneous algorithms across scientific domains.

cs.LG↗

arXiv2026

Agents' Last Exam

Recent AI systems have achieved strong results on a wide range of benchmarks, yet these gains have not translated into economically meaningful deployment across many professional domains. We argue that this gap is largely an evaluation problem: widely used benchmarks lack sustained performance measurement on real and economically valuable workflows. This paper introduces Agents' Last Exam (ALE), a benchmark designed to evaluate AI agents on long horizon, economically valuable, real world tasks with verifiable outcomes. Developed in collaboration with 250+ industry experts, ALE covers non-physical industries defined with reference to O*NET / SOC 2018 (the U.S. federal occupational taxonomy). It is organized around a task taxonomy with 55 sub fields grouped into 13 industry clusters covering 1K+ tasks. Current results show that the hardest tier remains far from saturated: across mainstream harness and backbone configurations, the average full pass rate is below 1%. ALE is designed as a living benchmark: its task pool grows continuously as new workflows and industries are onboarded. More broadly, ALE is intended not merely as another leaderboard, but as an instrument for closing the gap between benchmark success and GDP relevant impact.

cs.AI↗

arXiv2026

Quantifying the biophysical properties of stomatocytes in health and disease

Hereditary stomatocytosis (HS) comprises red blood cell (RBC) disorders characterized by cup-shaped erythrocytes that respond oppositely to splenectomy: curative in overhydrated HS (OHS) but potentially thrombogenic in dehydrated HS (DHS/xerocytosis). This paradox persists because RBC biomechanics is governed by partly independent parameters--shear modulus, bending rigidity, surface-to-volume ratio (S/V), and cytoplasmic viscosity--that existing assays capture only piecemeal. Here we combine dissipative particle dynamics (DPD) simulations with microfluidic imaging to construct a control discocyte and three stomatocyte models (ST-RBC1-3) at fixed membrane area and decreasing volume (109.7, 101.5, 89.8 fL), spanning the OHS-to-DHS range. Tracing this parameter set through five mechanically orthogonal assays, we find that interendothelial-slit (IES) traversal is geometry-dominated: overhydrated ST-RBC1 requires an order of magnitude higher critical pressure than healthy RBCs, whereas dehydrated ST-RBC3 passes freely. ST-RBC3 nonetheless suppresses membrane tank-treading and raises low-shear whole-blood viscosity by ~29% at physiological haematocrit, comparable to Gaucher-disease hyperviscosity. A funnel-obstacle chip amplifies these differences into a label-free centerline-offset signal predicted to separate all four RBC types (~4.5 standard deviations between extreme phenotypes). These results unite single-cell mechanics, splenic filtration, and hemorheology in one framework, resolve the splenectomy paradox, and point toward microfluidic pre-operative risk stratification in HS.

q-bio.CB↗

arXiv2026

Spectral Audit of In-Context Operator Networks

Existing evaluations of neural operators and in-context operator learning rely primarily on prediction error, but accurate output prediction does not guarantee the correct local dynamical structure. A model may match solutions while exhibiting incorrect sensitivities, distorted frequency response, spurious mode coupling, or unstable tangent behavior. We introduce a Jacobian-based spectral audit for in-context operator learning. For a fixed prompt, we differentiate the network output with respect to the query function and view the resulting Jacobian as a learned tangent operator. Projecting it onto Fourier modes, we obtain a local spectral characterization of the inferred operator, including frequency-dependent gains, phase structure, and cross-mode coupling. The audit complements standard prediction metrics by testing whether the model reproduces local mechanisms of the underlying PDE operator rather than only outputs. Across benchmarks, the audit reveals distinct operator-level phenomena, including phase transport, viscosity-dependent damping, nonlinear mode coupling, and reaction--diffusion stability structure. It also detects failures partially hidden by prediction-error metrics, including high-frequency degradation, incorrect phase recovery, and prompt--operator inconsistencies. Corrupted or internally inconsistent prompts lead to degraded tangent-operator structure even when pointwise predictions remain partially accurate. Our results suggest that prediction accuracy and local operator fidelity are distinct properties of learned neural operators. Our framework also provides a diagnostic for stability, sensitivity, and operator consistency.

math.NA↗

arXiv2026

Operator Learning for Reconstructing Flow Fields from Sparse Measurements: a Language Model Approach

Reconstructing flow fields from sparse measurements is a fundamental problem in fluid mechanics with broad implications for modeling, control, and design. In this work, we propose a novel operator learning framework that leverages the architecture of language models to perform flow reconstruction in a mesh-free manner. We reformulate flow field reconstruction as a sequence-to-sequence learning task, where sparse measurements are treated as context and unobserved locations as queries. Our model learns to reconstruct the full flow field from sparse inputs, effectively capturing spatial correlations and long-range dependencies. We evaluate the proposed approach on four benchmark datasets: (1) two-dimensional vortex street simulations, (2) daily average temperature data across the contiguous United States, (3) three-dimensional blood flow simulations based on dissipative particle dynamics, and (4) three-dimensional turbulent jet flow measurements obtained via particle tracking velocimetry. Across all cases, our method demonstrates competitive reconstruction accuracy, even with highly incomplete data (less than 10\% observed), and achieves efficient performance. The results highlight the potential of language models as robust and scalable tools for scientific data reconstruction, and suggest a promising direction toward the development of foundation models for scientific and engineering applications.

cs.CE↗

arXiv2026

Process-Informed Forecasting of Complex Thermal Dynamics in Pharmaceutical Manufacturing

Accurate time-series forecasting for complex physical systems is the backbone of modern industrial monitoring and control, yet deep learning models often lack the physical consistency required in regulated environments.To bridge this gap, we introduce Process-Informed Forecasting (PIF) models for temperature in pharmaceutical lyophilization, embedding deterministic production recipes as macro-structural priors. We investigate classical methods (e.g., Autoregressive Integrated Moving Average (ARIMA) model) and 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 to a new process. Our results show that PIF models outperform their data-driven counterparts in terms of accuracy, physical plausibility and noise resilience, offering a scalable framework for reliable and generalizable forecasting solutions in critical manufacturing.

cs.LG↗

arXiv2026

AMORE: Adaptive Multi-Output Operator Network for Stiff Chemical Kinetics

Time integration of stiff systems is a primary source of computational cost in combustion, hypersonics, and other reactive transport systems. This stiffness can introduce time scales significantly smaller than those associated with other physical processes, requiring extremely small time steps in explicit schemes or computationally intensive implicit methods. Consequently, strategies to alleviate challenges posed by stiffness are important. While neural operators (DeepONets) can act as surrogates for stiff kinetics, a reliable operator learning strategy is required to appropriately account for differences in error between output variables and samples. Here, we develop AMORE, Adaptive Multi-Output Operator Network, a framework comprising an operator capable of predicting multiple outputs and adaptive loss functions ensuring reliable operator learning. The operator predicts all thermochemical states from given initial conditions. We propose two adaptive loss functions within the framework, considering each state variable's and sample's error to penalize the loss function. We designed the trunk to automatically satisfy Partition of Unity. To enforce unity mass-fraction constraint exactly, we propose an invertible analytical map that transforms the $n$-dimensional species mass-fraction vector into an ($n-1$)-dimensional space. We extend the proposed adaptive loss functions to trunk and branch training in two-step training of DeepONet with multiple outputs. We implemented another unity mass fraction constraint exactly using a softmax function on the predicted mass fraction. We demonstrate efficacy and applicability of our models through two examples: syngas (12 states), GRI-Mech 3.0 (24 active states out of 54). The proposed DeepONet will be a backbone for future CFD studies to accelerate turbulent combustion simulations. AMORE is a general framework, and here, we also demonstrate it for FNO.

cs.LG↗