The Monte Carlo-type Neural Operator (MCNO) introduces a lightweight architecture for learning solution operators for parametric PDEs by directly approximating the kernel integral using a Monte Carlo approach. Unlike Fourier Neural Operators, MCNO makes no spectral or translation-invariance assumptions. The kernel is represented as a learnable tensor over a fixed set of randomly sampled points. This design enables generalization across multiple grid resolutions without relying on fixed global basis functions or repeated sampling during training. Experiments on standard 1D PDE benchmarks show that MCNO achieves competitive accuracy with low computational cost, providing a simple and practical alternative to spectral and graph-based neural operators.

Physics-informed neural operators offer a powerful framework for learning solution operators of partial differential equations (PDEs) by combining data and physics losses. However, these physics losses rely on derivatives. Computing these derivatives remains challenging, with spectral and finite difference methods introducing approximation errors due to finite resolution. Here, we propose the mollified graph neural operator ($m$GNO), the first method to leverage automatic differentiation and compute exact gradients on arbitrary geometries. This enhancement enables efficient training on irregular grids and varying geometries while allowing seamless evaluation of physics losses at randomly sampled points for improved generalization. For a PDE example on regular grids, $m$GNO paired with autograd reduced the L2 relative data error by 20x compared to finite differences, although training was slower. It can also solve PDEs on unstructured point clouds seamlessly, using physics losses only, at resolutions vastly lower than those needed for finite differences to be accurate enough. On these unstructured point clouds, $m$GNO leads to errors that are consistently 2 orders of magnitude lower than machine learning baselines (Meta-PDE, which accelerates PINNs) for comparable runtimes, and also delivers speedups from 1 to 3 orders of magnitude compared to the numerical solver for similar accuracy. $m$GNOs can also be used to solve inverse design and shape optimization problems on complex geometries.

Neural operators have emerged as a powerful, data-driven paradigm for learning solution operators of partial differential equations (PDEs). State-of-the-art architectures, such as the Fourier Neural Operator (FNO), have achieved remarkable success by performing convolutions in the frequency domain, making them highly effective for a wide range of problems. However, this method has some limitations, including the periodicity assumption of the Fourier transform. In addition, there are other methods of analysing a signal, beyond phase and amplitude perspective, and provide us with other useful information to learn an effective network. We introduce the \textbf{Hilbert Neural Operator (HNO)}, a new neural operator architecture to address some advantages by incorporating a strong inductive bias from signal processing. HNO operates by first mapping the input signal to its analytic representation via the Hilbert transform, thereby making instantaneous amplitude and phase information explicit features for the learning process. The core learnable operation -- a spectral convolution -- is then applied to this Hilbert-transformed representation. We hypothesize that this architecture enables HNO to model operators more effectively for causal, phase-sensitive, and non-stationary systems. We formalize the HNO architecture and provide the theoretical motivation for its design, rooted in analytic signal theory.

We investigate the high-precision training of Physics-Informed Neural Networks (PINNs) in unbounded domains, with a special focus on applications to singularity formulation in PDEs. We propose a modularized approach and study the choices of neural network ansatz, sampling strategy, and optimization algorithm. When combined with rigorous computer-assisted proofs and PDE analysis, the numerical solutions identified by PINNs, provided they are of high precision, can serve as a powerful tool for studying singularities in PDEs. For 1D Burgers equation, our framework can lead to a solution with very high precision, and for the 2D Boussinesq equation, which is directly related to the singularity formulation in 3D Euler and Navier-Stokes equations, we obtain a solution whose loss is $4$ digits smaller than that obtained in \cite{wang2023asymptotic} with fewer training steps. We also discuss potential directions for pushing towards machine precision for higher-dimensional problems.

Aircraft manufacturing is the jewel in the crown of industry, among which generating high-fidelity airfoil geometries with controllable and editable representations remains a fundamental challenge. While existing deep-learning-based methods rely on predefined parametric function families, e.g., B\'ezier curves and discrete point-based representations, they suffer from inherent trade-offs between expressiveness and resolution flexibility. To tackle this challenge, we introduce FuncGenFoil, a novel function-space generative model that directly learns functional airfoil geometries. Our method inherits both the advantages of arbitrary resolution sampling and the smoothness of parametric functions, as well as the strong expressiveness of discrete point-based functions. Empirical evaluations on the AFBench dataset demonstrate that FuncGenFoil improves upon state-of-the-art methods in airfoil generation by achieving a relative -74.4 label error reduction and +23.2 diversity increase on the AF-200K dataset. Our results highlight the advantages of function-space modeling for aerodynamic shape optimization, offering a powerful and flexible framework for high-fidelity airfoil design. Our code will be released.

We present NeuralOperator, an open-source Python library for operator learning. Neural operators generalize neural networks to maps between function spaces instead of finite-dimensional Euclidean spaces. They can be trained and inferenced on input and output functions given at various discretizations, satisfying a discretization convergence properties. Built on top of PyTorch, NeuralOperator provides all the tools for training and deploying neural operator models, as well as developing new ones, in a high-quality, tested, open-source package. It combines cutting-edge models and customizability with a gentle learning curve and simple user interface for newcomers.

Partial Differential Equations are foundational in modeling science and natural systems such as fluid dynamics and weather forecasting. The Latent Evolution of PDEs method is designed to address the computational intensity of classical and deep learning-based PDE solvers by proposing a scalable and efficient alternative. To enhance the efficiency and accuracy of LE-PDE, we incorporate the Mamba model, an advanced machine learning model known for its predictive efficiency and robustness in handling complex dynamic systems with a progressive learning strategy. The LE-PDE was tested on several benchmark problems. The method demonstrated a marked reduction in computational time compared to traditional solvers and standalone deep learning models while maintaining high accuracy in predicting system behavior over time. Our method doubles the inference speed compared to the LE-PDE while retaining the same level of parameter efficiency, making it well-suited for scenarios requiring long-term predictions.

Advancements in semiconductor fabrication over the past decade have catalyzed extensive research into all-optical devices driven by exciton-polariton condensates. Preliminary validations of such devices, including transistors, have shown encouraging results even under ambient conditions. A significant challenge still remains for large scale application however: the lack of a robust solver that can be used to simulate complex nonlinear systems which require an extended period of time to stabilize. Addressing this need, we propose the application of a machine-learning-based Fourier Neural Operator approach to find the solution to the Gross-Pitaevskii equations coupled with extra exciton rate equations. This work marks the first direct application of Neural Operators to an exciton-polariton condensate system. Our findings show that the proposed method can predict final-state solutions to a high degree of accuracy almost 1000 times faster than CUDA-based GPU solvers. Moreover, this paves the way for potential all-optical chip design workflows by integrating experimental data.