arXiv subjects

Omer San

Publications and source records attributed to Omer San.

At least 19 recordsLinked to original sources

arXiv2026

Two-Scale Localized PCA-Net: Coarse-Global and Local-Residual Representations for Artifact-Reduced PDE Operator Learning

Localized dimensionality reduction improves the scalability of operator learning for high-dimensional partial differential equations (PDEs), but independently decoded local patches can introduce block offsets, interface mismatches, and spurious high-wavenumber content. We introduce Two-Scale Localized PCA-Net, which decomposes the solution into a coarse-global component and local residual corrections. A compact global PCA basis captures domain-scale structure, while nonoverlapping local PCA bases represent the remaining fine-scale residual. A block-balanced latent objective couples the two representations, and optional interface-aware fine-tuning further promotes continuity through reconstruction and trace losses. On Poisson benchmarks, the two-scale representation substantially reduces reconstruction error and visible block artifacts relative to plain and overlap-based localized PCA-Net while approximately halving PCA fitting cost relative to overlap. On heterogeneous Darcy flow, it strongly reduces interface and discrete-residual errors, with more modest reconstruction gains. Ablations show that the primary improvement arises from the two-scale output representation, while interface-aware fine-tuning provides complementary continuity refinement. Overall, separating globally coherent structure from localized residual detail provides an efficient representation for artifact-reduced PDE operator learning.

cs.LG↗

arXiv2026

Spectral-Embedded Operator Learning for Three-Phase Interfacial Flow: A Ternary Cahn-Hilliard-Navier-Stokes Benchmark

Operator-learning surrogates have been benchmarked largely on single-field, single-interface problems, leaving unclear whether architectural choices validated in those settings transfer to constrained, multiphase flows. We introduce a three-phase interfacial-flow benchmark to examine whether the trunk coordinate representation matters for a multi-channel, interface-dominated target. The configuration consists of an air bubble rising through water, piercing a water-oil interface, and entraining a water plume into the oil within a bounded, wall-confined domain. Reference data are generated using a structure-preserving ternary Cahn-Hilliard-Navier-Stokes solver that algebraically preserves the simplex constraint. From 1,024 Sobol-sampled simulations spanning a nine-dimensional parameter space, we learn the mapping from physical parameters to five-channel space-time fields. We compare three parameter-matched DeepONet variants differing only in trunk representation: raw coordinates (DeepONet), random Fourier features (FEDONet), and a fixed tensor-product Chebyshev dictionary (SEDONet). SEDONet reduces the test relative L2 error by 16.8% compared with FEDONet and by 24.0% compared with DeepONet, while improving all five output channels. Spatial and temporal error analyses localize the principal gains near the diffuse interfaces and after bubble breakthrough. The results indicate that the Chebyshev representation is particularly effective for the strongly non-periodic wall-normal and temporal structure of this three-phase flow.

physics.flu-dyn↗

arXiv2026

Quantum SEDONet: Spectrally-Embedded Quantum Deep Operator Networks for Partial Differential Equations

Quantum DeepONet accelerates neural-operator inference by evaluating an orthogonally parameterized network on a quantum computer, reproducing in ideal simulation the accuracy of its classical counterpart at asymptotically lower inference cost. Its trunk network, however, receives query coordinates with limited spectral structure, requiring the network to learn oscillatory features through its nonlinearities. We propose Quantum SEDONet (Spectral-Embedded Deep Operator Network), which assigns each trunk coordinate a spectral basis according to its boundary condition: Fourier features for periodic coordinates and Chebyshev features for bounded, non-periodic coordinates. The basis is selected per coordinate rather than per problem, allowing both representations within a single problem. Under unary amplitude encoding, the embedding incurs no additional qubits or circuit depth when its dimension remains within the network width, while increasing the parameter count by only a few percent. Across four benchmarks, Quantum SEDONet reduces the mean relative L2 error by 54.1% for the antiderivative, 49.6% for advection, 36.0% for Burgers, and 36.2% for a mixed-boundary channel Poisson problem. Quantum and classical evaluation paths agree to within 10^-8 throughout. The channel Poisson problem simultaneously uses Fourier features in the periodic direction and Chebyshev features in the bounded direction, demonstrating coordinate-wise boundary-matched spectral embedding without additional quantum-resource cost.

quant-ph↗

arXiv2026

Iterative Refinement Diffusion for Super-Resolved Data Assimilation of Multiscale Physical Systems

Recovering high-resolution states from sparse, low-resolution observations is a central challenge in scientific machine learning and data assimilation. Classical data assimilation exploits temporal information through forecast-analysis cycles, but often requires repeated access to expensive high-resolution forecast models. Generative super-resolution can recover unresolved structure from coarse observations, but is commonly used as a one-shot mapping that does not fully exploit constraints from past states. We introduce Iterative Refinement (IR), a learned data assimilation framework that combines these perspectives. Instead of performing a single coarse-to-fine reconstruction, IR decomposes the task into resolution-wise forecast-analysis operations across a multiresolution hierarchy. At each stage, a shared neural operator with resolution-dependent spectral mode slicing provides a dynamical prior, while a shared conditional diffusion corrector uses the current coarser-resolution state to produce a refined posterior at the next finer resolution. We evaluate IR on one-dimensional stochastically forced Burgers dynamics and two-dimensional Kraichnan turbulence. On the challenging 256x256 Kraichnan benchmark, IR achieves an RMSE of 0.184 and an SSIM of 0.836, outperforming spectral upsampling, one-shot diffusion super-resolution, enhanced deep super-resolution, and an autoregressive forecaster. On the more constrained Burgers testbed, IR remains competitive with one-shot diffusion, which achieves the lowest RMSE. These results show that one-shot generative reconstruction can be effective for simpler settings, while hierarchical forecast-analysis refinement becomes advantageous in strongly multiscale and underdetermined regimes. Overall, IR combines temporal priors, generative correction, and multiresolution reconstruction for learned data assimilation in complex physical systems.

cs.LG↗

arXiv2026

Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximation

Deep Operator Networks (DeepONets) have emerged as a powerful framework for data-driven operator learning, providing flexible surrogates for nonlinear mappings arising in partial differential equations (PDEs). However, the standard trunk network, which operates directly on raw spatial or spatiotemporal coordinates through fully connected layers, often struggles to represent sharp gradients, boundary layers, and other non-periodic solution structures on bounded domains. To address these limitations, we introduce the Spectral-Embedded Deep Operator Network (SEDONet), a novel DeepONet architecture in which the trunk is driven by a fixed Chebyshev spectral dictionary instead of coordinate inputs. This non-periodic spectral embedding provides a principled inductive bias for bounded domains, enabling the learned operator to capture fine-scale features that are difficult for Fourier-based or MLP-only trunks to represent. SEDONet is evaluated on the 2-D Poisson equation, 1-D Burgers' equation, 1-D advection-diffusion equation, Allen-Cahn equation, Lorenz-96 chaotic system, and Darcy flow, covering elliptic, hyperbolic, parabolic, chaotic, and multiscale problems. Across all benchmarks, SEDONet consistently achieves the lowest or statistically comparable relative $L^2$ errors among DeepONet, FEDONet, and SEDONet, with improvements of up to 54% over the baseline DeepONet and consistent gains over Fourier-embedded variants on bounded, non-periodic problems. Energy spectrum analyses further demonstrate that SEDONet more accurately preserves intermediate- and high-frequency solution structures. The proposed framework provides a simple, parameter-neutral modification to DeepONets, offering a robust and computationally efficient spectral approach for surrogate modeling of nonlinear operators in scientific computing.

cs.LG↗

arXiv2026

Accelerating Bayesian inverse design in computational fluid dynamics using neural operators

Bayesian inverse design provides a principled framework for inferring aerodynamic geometries from sparse flow observations while quantifying uncertainty. However, its practical use in computational fluid dynamics (CFD) is severely limited by the cost of repeated high-fidelity simulations required for gradient-based Markov chain Monte Carlo (MCMC) sampling. While surrogate models are commonly proposed to reduce this cost, their effect on posterior geometry and uncertainty, especially for shock-dominated flows, remains poorly understood. In this work, we demonstrate that neural operator surrogates can be embedded directly within the MCMC inference loop while preserving posterior structure. Using a fully Bayesian inverse formulation of quasi-one-dimensional nozzle flow, we demonstrate that geometry parameterization plays a decisive role in identifiability and posterior conditioning, with cubic B-splines yielding stable and physically meaningful uncertainty estimates. Building on this formulation, a Deep Operator Network trained on CFD-generated data is substituted for the CFD solver within a No-U-Turn Sampler, while keeping the likelihood model, priors, and sampling configuration unchanged. Across sparse to fully observed regimes, surrogate-based inference reproduces the posterior geometry and uncertainty trends of the CFD reference. As a result of surrogate integration, total inference time is reduced to under one second, corresponding to a speedup exceeding three orders of magnitude. In addition, a direct inverse neural operator is examined as a deterministic alternative for inverse design, enabling single-shot geometry reconstruction without posterior sampling. These results demonstrate that neural operator-accelerated Bayesian inference enables practical, uncertainty-aware inverse design workflows for aerodynamic applications.

physics.flu-dyn↗

arXiv2026

The Evolution of Digital Twins from Reactive to Agentic Systems

Digital twins are evolving into self-learning, autonomous systems that link models, data, and human interaction. Realizing their full potential depends on interoperability, standardization, and the integration of artificial intelligence and advanced computational reasoning across sectors.

cs.CE↗

arXiv2026

WLNO: Wavelet-Laplace Neural Operator for Solving Partial Differential Equations

This work introduces the Wavelet-Laplace Neural Operator (WLNO), a novel neural operator that fuses Haar wavelet multi-scale spatial decomposition with the Laplace-domain pole-residue formulation of the Laplace Neural Operator (LNO). While LNO captures transient and steady-state dynamics through learnable system poles and residues, it lacks an explicit mechanism for extracting spatially localized multi-scale features inherent in complex PDE solutions. WLNO addresses this by augmenting the LNO core with a parallel single-level Haar discrete wavelet transform (DWT) branch that decomposes the lifted feature map into four frequency subbands: approximation (LL), horizontal detail (LH), vertical detail (HL), and diagonal detail (HH) and applies independent learned $1\times1$ convolutions to each subband before reconstruction via the inverse DWT. The two branches are fused through a learnable sigmoid-gated weight $α_\mathrm{wav}$, initialized to give a small initial contribution to the wavelet branch, allowing the model to adaptively balance Laplace-domain dynamics against spatial multi-scale features throughout training. WLNO is evaluated against LNO on five benchmark PDE problems using identical hyperparameters, training data, and evaluation protocols: the diffusion equation, the Burgers equation, the reaction-diffusion system, Darcy flow, and the two-dimensional Navier-Stokes equation. WLNO consistently outperforms LNO on all five problems, with the most pronounced improvement on problems with strong spatial multi-scale structure, such as the Burgers equation with sharp shock fronts and the Navier-Stokes equation with coherent vortical structures, while remaining consistent across smoother and elliptic problems. These results demonstrate that wavelet-based multi-scale spatial decomposition is a principled and effective complement to Laplace-domain operator learning.

cs.LG↗

arXiv2026

The impact of observation density on Bayesian inversion of latent dynamics in shock-dominated flows

Inferring unknown initial states in shock-dominated compressible flows from sparse and noisy measurements is a challenging ill-posed inverse problem due to nonlinear wave interactions and limited sensing. In this work, we develop a non-intrusive reduced-order modeling framework for efficient Bayesian initial-state inversion with uncertainty quantification. The framework combines a convolutional autoencoder with a learned latent-space forward operator. The autoencoder compresses high-dimensional flow fields into a compact nonlinear latent representation, while the forward operator predicts final-time latent states from encoded initial conditions. This AE-ROM surrogate enables rapid forward evaluations and is embedded within a No-U-Turn Sampler (NUTS) for posterior exploration. The framework is demonstrated using 500 high-fidelity Sod shock tube simulations generated through Latin hypercube sampling and solved using a fifth-order WENO scheme. The inverse problem seeks to recover unknown left and right density and pressure states from sparse noisy observations of final-time density and pressure fields. Results show that the AE-ROM accurately reconstructs key shock-tube structures, including the rarefaction wave, contact discontinuity, and shock front. A latent dimension of 32 provides an effective balance between reconstruction accuracy and reduced-space compactness, while 250 training simulations are sufficient for accurate reconstruction. Increasing observation density significantly contracts posterior uncertainty, reducing the mean posterior standard deviation by approximately 78% for density and 76% for pressure. Overall, the proposed framework provides a computationally efficient and uncertainty-aware approach for inverse analysis of shock-dominated flows, with potential extensions to multidimensional compressible-flow and digital-twin applications.

cs.LG↗

arXiv2026

Digital Twins in Coronary Artery Disease: A Mathematical Roadmap

The combination of data and models, enhanced by AI methodologies, leads to the paradigm called Digital Twins. This concept is expected to bring unprecedented support to personalized medicine. The combination of mathematical and numerical models with diagnostic devices that provide patient-specific knowledge in a bidirectional framework can be a formidable decision support for clinicians. In this paper, we consider some mathematical aspects of constructing a Digital Twin to prevent and treat Coronary Artery Disease. The keywords for the bidirectional communication between twins in our system are (i) Data Assimilation and (ii) Probabilistic Graphic Models. In particular, a quantity of paramount interest in the evaluation and prognosis of Coronary Artery Disease is the Wall Shear Stress, i.e., the tangential component of normal stress on the arterial wall. By considering steps for the personalization and the synthesis of Wall Shear Stress estimation, we propose a mathematical roadmap for constructing a Digital Twin system that could help prevent infarcts, one of the most lethal diseases in the world.

math.NA↗

arXiv2026

Hyperfastrl: Hypernetwork-based reinforcement learning for unified control of parametric chaotic PDEs

Spatiotemporal chaos in fluid systems exhibits severe parametric sensitivity, rendering classical adjoint-based optimal control intractable because each operating regime requires recomputing the control law. We address this bottleneck with hyperFastRL, a parameter-conditioned reinforcement learning framework that leverages Hypernetworks to shift from tuning isolated controllers per-regime to learning a unified parametric control manifold. By mapping a physical forcing parameter μ directly to the weights of a spatial feedback policy, the architecture cleanly decouples parametric adaptation from spatial boundary stabilization. To overcome the extreme variance inherent to chaotic reward landscapes, we deploy a pessimistic distributional value estimation over a massively parallel environment ensemble. We evaluate three Hypernetwork functional forms, ranging from residual MLPs to periodic Fourier and Kolmogorov-Arnold (KAN) representations, on the Kuramoto-Sivashinsky equation under varying spatial forcing. All forms achieve robust stabilization. KAN yields the most consistent energy-cascade suppression and tracking across unseen parametrizations, while Fourier networks exhibit worse extrapolation variability. Furthermore, leveraging high-throughput parallelization allows us to intentionally trade a fraction of peak asymptotic reward for a 37% reduction in training wall-clock time, identifying an optimal operating regime for practical deployment in complex, parameter-varying chaotic PDEs.

cs.CE↗

arXiv2026

SIMR-NO: A Spectrally-Informed Multi-Resolution Neural Operator for Turbulent Flow Super-Resolution

Reconstructing high-resolution turbulent flow fields from severely under-resolved observations is a fundamental inverse problem in computational fluid dynamics and scientific machine learning. Classical interpolation methods fail to recover missing fine-scale structures, while existing deep learning approaches rely on convolutional architectures that lack the spectral and multiscale inductive biases necessary for physically faithful reconstruction at large upscaling factors. We introduce the Spectrally-Informed Multi-Resolution Neural Operator (SIMR-NO), a hierarchical operator learning framework that factorizes the ill-posed inverse mapping across intermediate spatial resolutions, combines deterministic interpolation priors with spectrally gated Fourier residual corrections at each stage, and incorporates local refinement modules to recover fine-scale spatial features beyond the truncated Fourier basis. The proposed method is evaluated on Kolmogorov-forced two-dimensional turbulence, where $128\times128$ vorticity fields are reconstructed from extremely coarse $8\times8$ observations representing a $16\times$ downsampling factor. Across 201 independent test realizations, SIMR-NO achieves a mean relative $\ell_2$ error of $26.04\%$ with the lowest error variance among all methods, reducing reconstruction error by $31.7\%$ over FNO, $26.0\%$ over EDSR, and $9.3\%$ over LapSRN. Beyond pointwise accuracy, SIMR-NO is the only method that faithfully reproduces the ground-truth energy and enstrophy spectra across the full resolved wavenumber range, demonstrating physically consistent super-resolution of turbulent flow fields.

cs.LG↗

arXiv2026

FEDONet : Fourier-Embedded DeepONet for Spectrally Accurate Operator Learning

Deep Operator Networks (DeepONets) have recently emerged as powerful data-driven frameworks for learning nonlinear operators, particularly suited for approximating solutions to partial differential equations. Despite their promising capabilities, the standard implementation of DeepONets, which typically employs fully connected linear layers in the trunk network, can encounter limitations in capturing complex spatial structures inherent to various PDEs. To address this limitation, we use Fourier-Embedded trunk networks within the DeepONet architecture, leveraging random Fourier features to enrich spatial representation capabilities. The Fourier-Embedded DeepONet (FEDONet) demonstrates superior performance compared to the traditional DeepONet across a comprehensive suite of PDE-driven datasets, including the Burgers', 2D Poisson, Eikonal, Allen-Cahn, and the Kuramoto-Sivashinsky equation. To systematically evaluate the effectiveness of the architectures, we perform comparisons across multiple training dataset sizes and input noise levels. FEDONet delivers consistently superior reconstruction accuracy across all benchmark PDEs, with particularly large relative $L^2$ error reductions observed in chaotic and stiff systems. This work demonstrates the effectiveness of Fourier embeddings in enhancing neural operator learning, offering a robust and broadly applicable methodology for PDE surrogate modeling.

cs.LG↗

arXiv2026

Stabilizing autoregressive forecasts in chaotic systems via multi-rate latent recurrence

Long-horizon autoregressive forecasting of chaotic dynamical systems remains challenging due to rapid error amplification and distribution shift: small one-step inaccuracies compound into physically inconsistent rollouts and collapse of large-scale statistics. We introduce MSR-HINE, a hierarchical implicit forecaster that augments multiscale latent priors with multi-rate recurrent modules operating at distinct temporal scales. At each step, coarse-to-fine recurrent states generate latent priors, an implicit one-step predictor refines the state with multiscale latent injections, and a gated fusion with posterior latents enforces scale-consistent updates; a lightweight hidden-state correction further aligns recurrent memories with fused latents. The resulting architecture maintains long-term context on slow manifolds while preserving fast-scale variability, mitigating error accumulation in chaotic rollouts. Across two canonical benchmarks, MSR-HINE yields substantial gains over a U-Net autoregressive baseline: on Kuramoto-Sivashinsky it reduces end-horizon RMSE by 62.8% at H=400 and improves end-horizon ACC by +0.983 (from -0.155 to 0.828), extending the ACC >= 0.5 predictability horizon from 241 to 400 steps; on Lorenz-96 it reduces RMSE by 27.0% at H=100 and improves end horizon ACC by +0.402 (from 0.144 to 0.545), extending the ACC >= 0.5 horizon from 58 to 100 steps.

cs.LG↗

arXiv2025

Method of Manufactured Learning for Solver-free Training of Neural Operators

Training neural operators to approximate mappings between infinite-dimensional function spaces often requires extensive datasets generated by either demanding experimental setups or computationally expensive numerical solvers. This dependence on solver-based data limits scalability and constrains exploration across physical systems. Here we introduce the Method of Manufactured Learning (MML), a solver-independent framework for training neural operators using analytically constructed, physics-consistent datasets. Inspired by the classical method of manufactured solutions, MML replaces numerical data generation with functional synthesis, i.e., smooth candidate solutions are sampled from controlled analytical spaces, and the corresponding forcing fields are derived by direct application of the governing differential operators. During inference, setting these forcing terms to zero restores the original governing equations, allowing the trained neural operator to emulate the true solution operator of the system. The framework is agnostic to network architecture and can be integrated with any operator learning paradigm. In this paper, we employ Fourier neural operator as a representative example. Across canonical benchmarks including heat, advection, Burgers, and diffusion-reaction equations. MML achieves high spectral accuracy, low residual errors, and strong generalization to unseen conditions. By reframing data generation as a process of analytical synthesis, MML offers a scalable, solver-agnostic pathway toward constructing physically grounded neural operators that retain fidelity to governing laws without reliance on expensive numerical simulations or costly experimental data for training.

cs.LG↗

arXiv2025

Large Language Models for Control

This paper investigates using large language models (LLMs) to generate control actions directly, without requiring control-engineering expertise or hand-tuned algorithms. We implement several variants: (i) prompt-only, (ii) tool-assisted with access to historical data, and (iii) prediction-assisted using learned or simple models to score candidate actions. We compare them on tracking accuracy and actuation effort, with and without a prompt that requests lower actuator usage. Results show prompt-only LLMs already produce viable control, while tool-augmented versions adapt better to changing objectives but can be more sensitive to constraints, supporting LLM-in-the-loop control for evolving cyber-physical systems today and operator and human inputs.

eess.SY↗

arXiv2025

Hybrid Modeling, Sim-to-Real Reinforcement Learning, and Large Language Model Driven Control for Digital Twins

This work investigates the use of digital twins for dynamical system modeling and control, integrating physics-based, data-driven, and hybrid approaches with both traditional and AI-driven controllers. Using a miniature greenhouse as a test platform, four predictive models Linear, Physics-Based Modeling (PBM), Long Short Term Memory (LSTM), and Hybrid Analysis and Modeling (HAM) are developed and compared under interpolation and extrapolation scenarios. Three control strategies Model Predictive Control (MPC), Reinforcement Learning (RL), and Large Language Model (LLM) based control are also implemented to assess trade-offs in precision, adaptability, and implementation effort. Results show that in modeling HAM provides the most balanced performance across accuracy, generalization, and computational efficiency, while LSTM achieves high precision at greater resource cost. Among controllers, MPC delivers robust and predictable performance, RL demonstrates strong adaptability, and LLM-based controllers offer flexible human-AI interaction when coupled with predictive tools.

cs.AI↗

arXiv2025

Superresolving Non-linear PDE Dynamics with Reduced-Order Autodifferentiable Ensemble Kalman Filtering For Turbulence Modeling and Flow Regulation

Accurately reconstructing and forecasting high-resolution (HR) states from computationally cheap low-resolution (LR) observations is central to estimation-and-control of spatio-temporal PDE systems. We develop a unified superresolution pipeline based on the reduced-order autodifferentiable Ensemble Kalman filter (ROAD-EnKF). The method learns a low-dimensional latent dynamics model and a nonlinear decoder from latent variables to HR fields; the learned pair is embedded in an EnKF, enabling simultaneous state estimation and control-oriented forecasting with quantified uncertainty. We evaluate on three benchmarks: 1-D viscous Burgers equation (shock formation), Kuramoto-Sivashinsky (KS) equation (chaotic dynamics), and 2-D Navier-Stokes-Kraichnan turbulence (NSKT) (vortex decaying dynamics at Re 16,000). LR data are obtained by factors of 4-8 downsampling per spatial dimension and are corrupted with noise. On Burgers and KS, the latent models remain stable far beyond the observation window, accurately predicting shock propagation and chaotic attractor statistics up to 150 steps. On 2-D NSKT, the approach preserves the kinetic-energy spectrum and enstrophy budget of the HR data, indicating suitability for control scenarios that depend on fine-scale flow features. These results position ROAD-EnKF as a principled and efficient framework for physics-constrained superresolution, bridging LR sensing and HR actuation across diverse PDE regimes.

physics.flu-dyn↗