adaptive trunk deeponet

To address the limited expressivity and generalization capability of DeepONet in learning operators for two- and three-dimensional partial differential equations (PDEs), this paper proposes PoU-MoE DeepONet—a novel multi-trunk ensemble architecture that integrates partition-of-unity (PoU)-based sparse modeling with a spatial mixture-of-experts (MoE) mechanism, systematically augmented by basis enhancement and POD-guided spatial localization. We establish its universal approximation property theoretically. On multiple 2D/3D PDE operator learning benchmarks, PoU-MoE DeepONet achieves 2–4× lower relative ℓ₂ error compared to standard DeepONet and POD-DeepONet, significantly improving accuracy, robustness, and spatially localized representation. The framework offers a scalable and interpretable paradigm for learning complex physical operators.

Neural operators—deep models mapping between function spaces rather than vector spaces—lack open-source, discretization-agnostic implementations with theoretical convergence guarantees. To address this gap, we introduce NeuralOperator, the first modular and extensible Python library for neural operators built on PyTorch. It systematically supports state-of-the-art architectures—including Fourier Neural Operators (FNO) and Multipole Graph Neural Operators (MGNO)—and enables training and inference with functional inputs/outputs under diverse discretizations while rigorously ensuring discretization consistency and convergence. Through a unified API, comprehensive test coverage, and an end-to-end deployment toolchain, NeuralOperator significantly lowers the barrier to adopting neural operators in scientific computing tasks such as partial differential equation solving. The library bridges cutting-edge representational capacity with production-grade engineering robustness, making it both research-ready and deployable in real-world applications.

Scientific problems are often formulated in infinite-dimensional function spaces (e.g., PDE solution operators), whereas mainstream deep learning models are restricted to finite-dimensional mappings, limiting their generalization in scientific computing. To address this, we propose a systematic paradigm for extending classical neural networks (e.g., CNNs, Transformers) into *neural operators*, introducing— for the first time—the four fundamental design principles for function-space mappings, enabling low-intrusion, analytically tractable architectural migration. Our method integrates Fourier/wavelet-based operators, multi-scale attention, and discretization-invariance constraints, augmented by spectral-domain projection and mesh-agnostic parameterization. Evaluated on benchmarks including Navier–Stokes and Darcy flow equations, our models achieve substantial improvements in generalization across varying geometries, boundary conditions, and material coefficients; they also deliver 3.2× inference speedup and reduce generalization error by 47%.

To address the issues of excessive batch size, poor generalization, high memory consumption, and prolonged training time arising from uniform grid sampling in DeepONet training, this work proposes— for the first time—the integration of stochastic sampling directly into the trunk network’s input layer, replacing conventional fixed-grid sampling. In each training iteration, input points are dynamically sampled from varying spatial locations, thereby substantially reducing per-iteration batch size while enhancing robustness to functional distribution shifts. Theoretical analysis and experiments on three benchmark PDE tasks demonstrate that the proposed method achieves comparable or slightly improved test accuracy, yet reduces training time by 30–50% and GPU memory usage by 40–60%. Moreover, it significantly improves generalization capability and noise robustness. The core contribution lies in the first deep coupling of stochastic sampling with the DeepONet architecture, enabling simultaneous optimization of accuracy, computational efficiency, and generalizability.

Existing attention mechanisms operate on discrete sequences, limiting their applicability to continuous function spaces essential for scientific machine learning tasks such as PDE solving and physical simulation. Method: This work generalizes attention to continuous function spaces by introducing the Transformer Neural Operator (TNO), the first rigorously defined attention mechanism on functions. It establishes a mathematically sound formulation of functional attention and proposes a patching-based continuous attention mechanism coupled with an efficient discretization strategy to mitigate computational complexity in high dimensions. Contribution/Results: TNO is proven to be a universal approximator for arbitrary continuous operators. Experiments demonstrate that it significantly outperforms state-of-the-art neural operators across diverse PDE benchmarks and physics-informed simulation tasks, validating its effectiveness, scalability, and generalization capability in scientific machine learning.

This work addresses the challenge of reliable prediction and uncertainty quantification in function space mapping under sparse and irregular observations. The authors propose a unified encoder–decoder architecture that integrates neural processes with neural operators, introducing for the first time the conditional mechanism of neural processes into the neural operator framework to enable uncertainty-aware operator learning sensitive to local geometry. Two conditioning strategies—convolutional pooling summaries and query-aligned attention—are employed alongside stochastic latent variables and local geometric pathways, and the model is trained on both function regression and partial differential equation (PDE) tasks. Experiments demonstrate that the method achieves efficient learning under sparse conditions across multiple PDE benchmarks, matching the performance of dense-grid methods in specific settings and highlighting the critical role of preserving local contextual geometry, particularly in non-periodic domains.

Modeling cross-dimensional physical quantities and their conservation laws on unstructured meshes with complex geometries remains highly challenging. This work proposes Topological Neural Operators (TNOs), which generalize neural operators to cell complexes for the first time by integrating discrete exterior calculus to explicitly model couplings among cells of varying dimensions through learnable gradient, curl, and divergence operators. TNOs decouple information propagation pathways from transformation mechanisms and incorporate hierarchically coarsened complexes to capture long-range dependencies and global topological structure. Experiments demonstrate that the method significantly outperforms existing approaches across multiple PDE benchmarks, including fluid dynamics problems on irregular geometries, thereby validating the efficacy of high-rank representations and native topological modeling.

This work addresses the limited out-of-distribution (OOD) generalization of existing neural operators, which typically require fine-tuning or retraining. The authors propose the Chain of Operators (CHOP) framework, which, for the first time, adapts prompt engineering concepts from large language models to the neural operator setting. CHOP constructs operator chains composed of explicit elementary transformations and a frozen In-Context Operator Network (ICON), enabling training-free OOD generalization without updating any parameters. The approach yields both interpretability and closed-form expressions, and demonstrates strong generalization capabilities—significantly reducing relative inference errors across diverse tasks, including scalar conservation laws, mean-field control problems, and cross-family partial differential equation settings.