In this section, we derive the variational updates. The last term contains cross entropies between prior and variational PG distributions. The full VB model affects this limit in two ways. Its impact on the second parameter is indirect. How plausible is this "good fit" regime?

We will first replicate an equation similar to (20) for the backward case. The derivation is similar to that of the forward equation, so that it uses a combination of equations (16), (18) and (19) while leaving out the observation likelihood function. The combination is again carried out using the Laplace transform.

A critical issue that affects engineers trying to assess the structural integrity of various infrastructures, such as metal rods or acoustic ducts, is the challenge of detecting internal fractures (defects). Traditionally, engineers depend on audible and visual aids to identify these fractures, as they do not physically dissect the object in question into multiple pieces to check for inconsistencies. This research introduces ideas towards the development of a robust strategy to image such defects using only a small set of minimal, non-invasive measurements. Assuming a one dimensional model (e.g. longitudinal waves in long and thin rods/acoustic ducts or transverse vibrations of strings), we make use of the continuous one-dimensional wave equation to model these physical phenomena and then employ specialized mathematical analysis tools (the Laplace transform and optimization) to introduce our defect imaging ideas. In particular, we will focus on the case of a long bar which is homogeneous throughout except in a small area where a defect in its Young's modulus is present. We will first demonstrate how the problem is equivalent to a spring-mass vibrational system, and then show how our imaging strategy makes use of the Laplace domain analytic map between the characteristics of the respective defect and the measurement data. More explicitly, we will utilize MATLAB (a platform for numerical computations) to collect synthetic data (computational alternative to real world measurements) for several scenarios with one defect of arbitrary location and stiffness. Subsequently, we will use this data along with our analytically developed map (between defect characteristics and measurements) to construct a residual function which, once optimized, will reveal the location and magnitude of the stiffness defect.

Modern optimization faces a fundamental challenge: local gradient-based methods provide no global information about the objective function $L$ landscape, often leading to suboptimal convergence and sensitivity to initialization. We introduce a novel optimization framework that leverages resurgence theory from complex analysis to extract global structural information from divergent asymptotic series. Our key insight is that the factorially divergent perturbative expansions of parameter space partition functions encode precise information about all critical objective function value in the landscape through their Borel transform singularities. The algorithm works by computing the statistical mechanical partition function $Z(g) = \int e^{-L(θ)/g} dθ$ for small coupling $g\ll 1$, extracting its asymptotic series coefficients, and identifying Borel plane singularities that correspond one-to-one with critical objective function values. These target values provide global guidance to local optimizers, enabling principled learning rate adaptation and escape from suboptimal regions. Unlike heuristic adaptive methods, targets are theoretically grounded in the geometry of the optimization landscape.

In this section, we derive the variational updates. The last term contains cross entropies between prior and variational PG distributions. The full VB model affects this limit in two ways. Its impact on the second parameter is indirect. How plausible is this "good fit" regime?

We will also expand Section 2 to make it more gradual, stating theorems and definitions explicitly.

We will first replicate an equation similar to (20) for the backward case. The derivation is similar to that of the forward equation, so that it uses a combination of equations (16), (18) and (19) while leaving out the observation likelihood function. The combination is again carried out using the Laplace transform.

-- Time series prediction, a crucial task across various domains, faces significant challenges due to the inherent complexities of time series data, including non-stationarity, multi-scale periodicity, and transient dynamics, particularly when tackling long-term predictions. While Transformer-based architectures have shown promise, their quadratic complexity with sequence length hinders their efficiency for long-term predictions. Meanwhile, they are susceptible to data noise issues in time series. This paper proposes a novel framework, FLDmamba (Fourier and Laplace Transform Decomposition Mamba), addressing these limitations. FLDmamba leverages the strengths of both Fourier and Laplace transforms to effectively capture both multi-scale periodicity, transient dynamics within time series data, and improve the robustness of the model to the data noise issue. Our extensive experiments demonstrate that FLDmamba achieves superior performance on time series prediction benchmarks, outperforming both Transformer-based and other Mamba-based architectures. IME series prediction, which forecasts the future values of a (multivariate) variable based on its historical values, finds its application across a wide range of fields. Examples include weather prediction [1, 2], power grid management [3], traffic prediction [4, 5, 6, 7, 8, 9, 10], and stock market [11, 12, 13, 14], to name just a few. Despite significant advancements in this domain, the inherent complexities of time series data, such as non-stationarity, multi-scale periodicity, intrinsic stochasticity, and noise, pose substantial challenges to existing predictive models in long-term prediction. Q. Zhang and S.M. Yiu are from the University of Hong Kong.

We introduce a frequency-domain framework for convergence analysis of hyperparameters in game optimization, leveraging High-Resolution Differential Equations (HRDEs) and Laplace transforms. Focusing on the Lookahead algorithm--characterized by gradient steps $k$ and averaging coefficient $α$--we transform the discrete-time oscillatory dynamics of bilinear games into the frequency domain to derive precise convergence criteria. Our higher-precision $O(γ^2)$-HRDE models yield tighter criteria, while our first-order $O(γ)$-HRDE models offer practical guidance by prioritizing actionable hyperparameter tuning over complex closed-form solutions. Empirical validation in discrete-time settings demonstrates the effectiveness of our approach, which may further extend to locally linear operators, offering a scalable framework for selecting hyperparameters for learning in games.

Modelling forced dynamical systems - where an external input drives the system state - is critical across diverse domains such as engineering, finance, and the natural sciences. In this work, we propose Laplace-Net, a decoupled, solver-free neural framework for learning forced and delay-aware systems. It leverages a Laplace transform-based approach to decompose internal dynamics, external inputs, and initial values into established theoretical concepts, enhancing interpretability. Laplace-Net promotes transferability since the system can be rapidly re-trained or fine-tuned for new forcing signals, providing flexibility in applications ranging from controller adaptation to long-horizon forecasting. Experimental results on eight benchmark datasets - including linear, non-linear, and delayed systems - demonstrate the method's improved accuracy and robustness compared to state-of-the-art approaches, particularly in handling complex and previously unseen inputs.