We introduce the Sumudu Neural Operator (SNO), a neural operator rooted in the properties of the Sumudu Transform. We leverage the relationship between the polynomial expansions of transform pairs to decompose the input space as coefficients, which are then transformed into the Sumudu Space, where the neural operator is parameterized. We evaluate the operator in ODEs (Duffing Oscillator, Lorenz System, and Driven Pendulum) and PDEs (Euler-Bernoulli Beam, Burger's Equation, Diffusion, Diffusion-Reaction, and Brus-selator). SNO achieves superior performance to FNO on PDEs and demonstrates competitive accuracy with LNO on several PDE tasks, including the lowest error on the Euler-Bernoulli Beam and Diffusion Equation. Additionally, we apply zero-shot super-resolution to the PDE tasks to observe the model's capability of obtaining higher quality data from low-quality samples. These preliminary findings suggest promise for the Sumudu Transform as a neural operator design, particularly for certain classes of PDEs.

Fast and accurate simulation of dynamical systems is a fundamental challenge across scientific and engineering domains. Traditional numerical integrators often face a trade-off between accuracy and computational efficiency, while existing neural network-based approaches typically require training a separate model for each case. To overcome these limitations, we introduce a novel multi-modal foundation model for large-scale simulations of differential equations: FMint-SDE (Foundation Model based on Initialization for stochastic differential equations). Based on a decoder-only transformer with in-context learning, FMint-SDE leverages numerical and textual modalities to learn a universal error-correction scheme. It is trained using prompted sequences of coarse solutions generated by conventional solvers, enabling broad generalization across diverse systems. We evaluate our models on a suite of challenging SDE benchmarks spanning applications in molecular dynamics, mechanical systems, finance, and biology. Experimental results show that our approach achieves a superior accuracy-efficiency tradeoff compared to classical solvers, underscoring the potential of FMint-SDE as a general-purpose simulation tool for dynamical systems.

ABSTRACT It was recently shown that the loss function used for training physics-informed neural networks (PINNs) exhibits local minima at solutions corresponding to fixed points of dynamical systems. In the forward setting, where the PINN is trained to solve initial value problems, these local minima can interfere with training and potentially leading to physically incorrect solutions. Building on stability theory, this paper proposes a regularization scheme that penalizes solutions corresponding to unstable fixed points. Experimental results on four dynamical systems, including the Lotka-V olterra model and the van der Pol oscillator, show that our scheme helps avoiding physically incorrect solutions and substantially improves the training success rate of PINNs. Index T erms-- PINNs, regularization, stability 1. INTRODUCTION Physics-informed neural networks (PINNs, [1]) are among the most prominent instantiations of physics-informed machine learning.

-- While accurate, black-box system identification models lack interpretability of the underlying system dynamics. This paper proposes State-Space Kolmogorov-Arnold Networks (SS-KAN) to address this challenge by integrating Kolmogorov-Arnold Networks within a state-space framework. The proposed model is validated on two benchmark systems: the Silverbox and the Wiener-Hammerstein benchmarks. Results show that SS-KAN provides enhanced interpretability due to sparsity-promoting regularization and the direct visualization of its learned univariate functions, which reveal system nonlinearities at the cost of accuracy when compared to state-of-the-art black-box models, highlighting SS-KAN as a promising approach for interpretable nonlinear system identification, balancing accuracy and interpretability of nonlinear system dynamics. YSTEM identification, the process of building mathematical models from observed data, is a fundamental discipline in engineering and control. Accurate system models are useful for tasks ranging from controller design and performance optimization to fault detection and system analysis.

Laplace Neural Operators (LNOs) have recently emerged as a promising approach in scientific machine learning due to the ability to learn nonlinear maps between functional spaces. However, this framework often requires substantial amounts of high-fidelity (HF) training data, which is often prohibitively expensive to acquire. To address this, we propose multi-fidelity Laplace Neural Operators (MF-LNOs), which combine a low-fidelity (LF) base model with parallel linear/nonlinear HF correctors and dynamic inter-fidelity weighting. This allows us to exploit correlations between LF and HF datasets and achieve accurate inference of quantities of interest even with sparse HF data. We further incorporate a modified replica exchange stochastic gradient Langevin algorithm, which enables a more effective posterior distribution estimation and uncertainty quantification in model predictions. Extensive validation across four canonical dynamical systems (the Lorenz system, Duffing oscillator, Burgers equation, and Brusselator reaction-diffusion system) demonstrates the framework's effectiveness. The results show significant improvements, with testing losses reduced by 40% to 80% compared to traditional approaches. This validates MF-LNO as a versatile tool for surrogate modeling in parametric PDEs, offering significant improvements in data efficiency and uncertainty-aware prediction.

Dynamical systems theory has long provided a foundation for understanding evolving phenomena across scientific domains. Yet, the application of this theory to complex real-world systems remains challenging due to issues in mathematical modeling, nonlinearity, and high dimensionality. In this work, we introduce a data-driven computational framework to derive low-dimensional linear models for nonlinear dynamical systems directly from raw experimental data. This framework enables global stability analysis through interpretable linear models that capture the underlying system structure. Our approach employs time-delay embedding, physics-informed deep autoencoders, and annealing-based regularization to identify novel low-dimensional coordinate representations, unlocking insights across a variety of simulated and previously unstudied experimental dynamical systems. These new coordinate representations enable accurate long-horizon predictions and automatic identification of intricate invariant sets while providing empirical stability guarantees. Our method offers a promising pathway to analyze complex dynamical behaviors across fields such as physics, climate science, and engineering, with broad implications for understanding nonlinear systems in the real world.

In computational engineering, enhancing the simulation speed and efficiency is a perpetual goal. To fully take advantage of neural network techniques and hardware, we present the SLiding-window Initially-truncated Dynamic-response Estimator (SLIDE), a deep learning-based method designed to estimate output sequences of mechanical or multibody systems with primarily, but not exclusively, forced excitation. A key advantage of SLIDE is its ability to estimate the dynamic response of damped systems without requiring the full system state, making it particularly effective for flexible multibody systems. The method truncates the output window based on the decay of initial effects, such as damping, which is approximated by the complex eigenvalues of the systems linearized equations. In addition, a second neural network is trained to provide an error estimation, further enhancing the methods applicability. The method is applied to a diverse selection of systems, including the Duffing oscillator, a flexible slider-crank system, and an industrial 6R manipulator, mounted on a flexible socket. Our results demonstrate significant speedups from the simulation up to several millions, exceeding real-time performance substantially.

Physically implemented neural networks hold the potential to achieve the performance of deep learning models by exploiting the innate physical properties of devices as computational tools. This exploration of physical processes for computation requires to also consider their intrinsic dynamics, which can serve as valuable resources to process information. However, existing computational methods are unable to extend the success of deep learning techniques to parameters influencing device dynamics, which often lack a precise mathematical description. In this work, we formulate a universal framework to optimise interactions with dynamic physical systems in a fully data-driven fashion. The framework adopts neural stochastic differential equations as differentiable digital twins, effectively capturing both deterministic and stochastic behaviours of devices. Employing differentiation through the trained models provides the essential mathematical estimates for optimizing a physical neural network, harnessing the intrinsic temporal computation abilities of its physical nodes. To accurately model real devices' behaviours, we formulated neural-SDE variants that can operate under a variety of experimental settings. Our work demonstrates the framework's applicability through simulations and physical implementations of interacting dynamic devices, while highlighting the importance of accurately capturing system stochasticity for the successful deployment of a physically defined neural network.