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.

Neuro-Symbolic Control with Large Language Models for Language-Guided Spatial Tasks

Although large language models (LLMs) have recently become effective tools for language-conditioned control in embodied systems, instability, slow convergence, and hallucinated actions continue to limit their direct application to continuous control. A modular neuro-symbolic control framework that clearly distinguishes between low-level motion execution and high-level semantic reasoning is proposed in this work. While a lightweight neural delta controller performs bounded, incremental actions in continuous space, a locally deployed LLM interprets symbolic tasks. We assess the suggested method in a planar manipulation setting with spatial relations between objects specified by language. Numerous tasks and local language models, such as Mistral, Phi, and LLaMA-3.2, are used in extensive experiments to compare LLM-only control, neural-only control, and the suggested LLM+DL framework. In comparison to LLM-only baselines, the results show that the neuro-symbolic integration consistently increases both success rate and efficiency, achieving average step reductions exceeding 70% and speedups of up to 8.83x while remaining robust to language model quality. The suggested framework enhances interpretability, stability, and generalization without any need of reinforcement learning or costly rollouts by controlling the LLM to symbolic outputs and allocating uninterpreted execution to a neural controller trained on artificial geometric data. These outputs show empirically that neuro-symbolic decomposition offers a scalable and principled way to integrate language understanding with ongoing control, this approach promotes the creation of dependable and effective language-guided embodied systems.

Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximation

Deep Operator Networks (DeepONets) have become a central tool in data-driven operator learning, providing flexible surrogates for nonlinear mappings arising in partial differential equations (PDEs). However, the standard trunk design based on fully connected layers acting on raw spatial or spatiotemporal coordinates struggles to represent sharp gradients, boundary layers, and non-periodic structures commonly found in PDEs posed on bounded domains with Dirichlet or Neumann boundary conditions. To address these limitations, we introduce the Spectral-Embedded DeepONet (SEDONet), a new DeepONet variant in which the trunk is driven by a fixed Chebyshev spectral dictionary rather than coordinate inputs. This non-periodic spectral embedding provides a principled inductive bias tailored to bounded domains, enabling the learned operator to capture fine-scale non-periodic features that are difficult for Fourier or MLP trunks to represent. SEDONet is evaluated on a suite of PDE benchmarks including 2D Poisson, 1D Burgers, 1D advection-diffusion, Allen-Cahn dynamics, and the Lorenz-96 chaotic system, covering elliptic, parabolic, advective, and multiscale temporal phenomena, all of which can be viewed as canonical problems in computational mechanics. Across all datasets, SEDONet consistently achieves the lowest relative L2 errors among DeepONet, FEDONet, and SEDONet, with average improvements of about 30-40% over the baseline DeepONet and meaningful gains over Fourier-embedded variants on non-periodic geometries. Spectral analyses further show that SEDONet more accurately preserves high-frequency and boundary-localized features, demonstrating the value of Chebyshev embeddings in non-periodic operator learning. The proposed architecture offers a simple, parameter-neutral modification to DeepONets, delivering a robust and efficient spectral framework for surrogate modeling of PDEs on bounded domains.