The Painlevé analysis and computational technique for new wave solutions with its numerical validation to the complex short pulse equation

Abstract We designed a new artificial neural network called Exposed latent state neural ordinary differential equation with physics (ExpNODE-p) by modifying the neural ordinary differential equation (NODE) framework to successfully predict the time evolution of the two-dimensional mode profile in nonlinear saturated stage. Starting from the magnetohydrodynamic equations, simplifying assumptions were applied based on physical properties and symmetry considerations of the energetic-particle-driven geodesic acoustic mode (EGAM) to reduce complexity. Our approach embeds known physical characteristics directly into the neural network architecture by exposing latent differential states, enabling the model to capture complex features in the nonlinear saturated stage that are difficult to describe analytically. ExpNODE-p was evaluated using a dataset generated from first-principles simulations of the EGAM instability, focusing on the nonlinear saturated stage where the mode properties (e.g. frequency) are quite difficult to capture. Compared to state-of-the-art models such as ConvLSTM, ExpNODE-p achieved superior performance in both accuracy and training efficiency for multi-step predictions. Additionally, the model exhibited strong generalization capabilities, accurately predicting mode profiles outside the training dataset and capturing detailed features and asymmetries inherent in the EGAM dynamics. Our results establish ExpNODE-p as a powerful tool for creating fast, accurate surrogate models of complex plasma phenomena, opening the door to applications that are computationally intractable with first-principles simulations.

Modeling Still Matters: A Surprising Instance of Catastrophic Floating Point Errors in Mathematical Biology and Numerical Methods for ODEs

A note on the local behavior of the Taylor method for stiff ODEs

Abstract Perovskite colloidal nanocrystals (PeNCs) exhibit outstanding optoelectronic properties, making them promising candidates for light‐emitting diodes (LEDs). Despite the remarkable performance improvements in perovskite‐based LEDs (PeLEDs) over the years, a comprehensive life cycle assessment (LCA), covering their synthesis and purification, operation, and end‐of‐life disposal, remains crucial as large‐scale production and commercialization approach. This study provides an exhaustive study of environmental impacts and costs on the fabrication of CsPbBr3 PeNCs by the conventional hot injection (HI) route and the alternative microwave‐assisted (MW) approach. Morphological, structural, and optical characterization confirms the high quality of both sets of NCs, although minor differences are detected. The MW route significantly reduces impacts at the laboratory scale due to its superior energy efficiency. However, at the industrial scale, both routes exhibit similar energy efficiencies, making the environmental comparison less conclusive. Additionally, n‐dodecane (DOD) is explored as an alternative solvent for PeNC synthesis, with its recovery via distillation successfully demonstrated. While DOD recovery reduces solvent consumption, the lower reaction yield results in a higher overall environmental impact. The MW ODE method currently offers the lowest total costs, primarily due to reduced labor expenses. Advancing the MW method with DOD to achieve yields comparable to HI could represent a breakthrough for sustainable PeNCs synthesis.

Approximate well-balanced WENO finite difference schemes using a global-flux quadrature method with multi-step ODE integrator weights

This research presents innovative modified explicit block methods with fifth-order algebraic accuracy to address initial value problems (IVPs). The derivation of the methods employs fitting coefficients that eliminate phase lag and amplification error, as well as their derivatives. A thorough stability analysis of the new approach is conducted. Comparative assessments with existing methods highlight the superior effectiveness of the proposed algorithms. Numerical tests verify that this technique significantly surpasses conventional methods for solving IVPs, particularly those exhibiting oscillatory solutions.

This paper investigates the pricing of basket credit default swaps (CDS) under stochastic interest rates using a reduced-form model. We assume the default intensity of reference entities and stochastic interest rates both follow Vasicek processes, with risk-free counterparties. Through PDE and ODE methods, we derive approximate closed-form solutions for the joint survival probability density and the probability density of first-default events among reference entities.

Study’s Excerpt:• The study evaluates numerical methods for ODEs to guide suitable model simulations.• Adomian Decomposition suits decay/growth models; block method excels in all problems, including SIR.• The study highlights numerical methods' ease and precision in simulating mathematical models.Full Abstract:This study presents a comparative analysis on some numerical methods for simulating mathematical models of ordinary differential equations, including Euler, Classical Runge-Kutta, Adomian Decomposition, Block, and Simulink. We examine each method's accuracy, stability, and consistency through a series of test cases. These methods are applied to simulate some selected mathematical models, and the results are shown in tables. The graph of each table is depicted in figures for discussion and comparative analysis. The results show that, while straightforward, the Euler method demonstrates significant limitations in accuracy compared to the Classical Runge-Kutta method, which provides reliable and precise results. The Adomian Decomposition Method solves the problems and yields results very close to the analytical solution, but the block method performs better due to its multistep approach. Simulink offers a more robust approach for modelling and simulation with visible and interpretable solutions for good understanding. This study revealed that numerical methods can easily be used to better simulate mathematical models that may not have analytical solutions and thus provide approximate solutions.

FxTS-Net: Fixed-time stable learning framework for Neural ODEs.

The paper concerns convergence and asymptotic statistics for stochastic approximation driven by Markovian noise: $$ \theta_{n+1}= \theta_n + \alpha_{n + 1} f(\theta_n, \Phi_{n+1}) \,,\quad n\ge 0, $$ in which each $\theta_n\in\Re^d$, $ \{ \Phi_n \}$ is a Markov chain on a general state space X with stationary distribution $\pi$, and $f:\Re^d\times \text{X} \to\Re^d$. In addition to standard Lipschitz bounds on $f$, and conditions on the vanishing step-size sequence $\{\alpha_n\}$, it is assumed that the associated ODE is globally asymptotically stable with stationary point denoted $\theta^*$, where $\bar f(\theta)=E[f(\theta,\Phi)]$ with $\Phi\sim\pi$. Moreover, the ODE@$\infty$ defined with respect to the vector field, $$ \bar f_\infty(\theta):= \lim_{r\to\infty} r^{-1} \bar f(r\theta) \,,\qquad \theta\in\Re^d, $$ is asymptotically stable. The main contributions are summarized as follows: (i) The sequence $\theta$ is convergent if $\Phi$ is geometrically ergodic, and subject to compatible bounds on $f$. The remaining results are established under a stronger assumption on the Markov chain: A slightly weaker version of the Donsker-Varadhan Lyapunov drift condition known as (DV3). (ii) A Lyapunov function is constructed for the joint process $\{\theta_n,\Phi_n\}$ that implies convergence of $\{ \theta_n\}$ in $L_4$. (iii) A functional CLT is established, as well as the usual one-dimensional CLT for the normalized error $z_n:= (\theta_n-\theta^*)/\sqrt{\alpha_n}$. Moment bounds combined with the CLT imply convergence of the normalized covariance, $$ \lim_{n \to \infty} E [ z_n z_n^T ] = \Sigma_\theta, $$ where $\Sigma_\theta$ is the asymptotic covariance appearing in the CLT. (iv) An example is provided where the Markov chain $\Phi$ is geometrically ergodic but it does not satisfy (DV3). While the algorithm is convergent, the second moment is unbounded.

ABSTRACTReal‐world systems are often formulated as constrained optimization problems. Techniques to incorporate constraints into neural networks (NN), such as neural ordinary differential equations (Neural ODEs), have been used. However, these introduce hyperparameters that require manual tuning through trial and error, raising doubts about the successful incorporation of constraints into the generated model. This paper describes in detail the two‐stage training method for Neural ODEs, a simple, effective, and penalty parameter‐free approach to model constrained systems. In this approach, the constrained optimization problem is rewritten as two optimization subproblems that are solved in two stages. The first stage aims at finding feasible NN parameters by minimizing a measure of constraints violation. The second stage aims to find the optimal NN parameters by minimizing the loss function while keeping inside the feasible region. We experimentally demonstrate that our method produces models that satisfy the constraints and also improves their predictive performance, thus ensuring compliance with critical system properties and also contributing to reducing data quantity requirements. Furthermore, we show that the proposed method improves the convergence to an optimal solution and improves the explainability of Neural ODE models. Our proposed two‐stage training method can be used with any NN architectures.

Variable stepsize general linear methods for ODEs

In this paper, we construct explicit second derivative general linear methods (SGLMs) with quadratic stability and a large region of absolute stability for the numerical solution of non-stiff ODEs. The methods are constructed in two different cases: SGLMs with p = q = r = s and SGLMs with p = q and r = s = 2 in which p, q, r and s are respectively the order, stage order, the number of external stages and the number of internal stages. Examples of the methods up to order five are given. The efficiency of the constructed methods is illustrated by applying them to some well-known non-stiff problems and comparing the obtained results with those of general linear methods of the same order and stage order.

There are several common procedures used to numerically integrate second-order ordinary differential equations. The most common one is to reduce the equation’s order by duplicating the number of variables. This allows one to take advantage of the family of Runge–Kutta methods or the Adams family of multi-step methods. Another approach is the use of methods that have been developed to directly integrate an ordinary differential equation without increasing the number of variables. An important drawback when using Runge–Kutta methods is that when one tries to apply them to differential algebraic equations, they require a reduction in the index, leading to a need for stabilization methods to remove the drift. In this paper, a new family of methods for the direct integration of second-order ordinary differential equations is presented. These methods can be considered as a generalization of the central differences method. The methods are classified according to the number of derivatives they take into account (degree). They include some parameters that can be chosen to configure the equation’s behavior. Some sets of parameters were studied, and some examples belonging to structural dynamics and multibody dynamics are presented. An example of the application of the method to a differential algebraic equation is also included.

Abstract We explain the relation between the r = 1 case of the logistic map x i+1 = r x i (1 − x i ), x i ∈ R , i = 0, 1, 2, …, r > 0 and x 0 ≥ 0, and the renormalization group flow arising in the multiscale analysis of interesting zero fixed point, asymptotic free quantum field theory models such as the ultraviolet (1 + 1)-dimensional Gross-Neveu model and QCD, and the infrared ϕ 4 4 model . We obtain the asymptotics of the mapping, which shows an inverse power decay approach to the fixed point x * = 0, Gaussian fixed point, with additional logarithmic-like corrections. This asymptotic behavior is independent of the initial condition x 0 ∈ (0, 1) (hence, there is no constraint for x 0 to be small, as usual in quantum field models), and only depends on the lowest orders in a polynomial perturbation. In asymptotic free quantum field theory, this amounts to say that knowing the renormalization group β-function expansion in the coupling constant, up to higher orders, does not improve our knowledge of the asymptotics of the coupling flow. A comparison with a similar differential equation with continuous time is made by analyzing stability of this kind of solution and higher order monomial perturbations. We also obtain the detailed asymptotics for 0 < r < 1. As well, our methods can be applied when r ∈ (1, 3]. It is known, but without detailed asymptotics, that all trajectories with initial condition x 0 ∈ ( − 1, 1) converge to the fixed point x * = (r − 1)/r. For r = 2, the super attractive case, we obtain an explicit exact solution which exhibits an exploding, non-constant exponential decay rate approach to the x * = (1/2) fixed point. Our methods include the use of iterations, a discrete version of the Fundamental theorem of Calculus, a discrete version of the integrating factor method for first order linear ODEs and, sometimes, a scaling transformation. To obtain these results, we do not use the traditional Banach contraction mapping theorem, which only provides an upper bound on the asymptotics. We expect that our methods can be employed to determine the asymptotics of the logistic map for a wider range of parameters, where other fixed points are present.

SMS: Spiking marching scheme for efficient long time integration of differential equations

In this study, we present a novel approach for the numerical solution of high-order ODEs and MTVOFDEs with BCs. Our method leverages a class of GSJPs that possess the crucial property of satisfying the given BCs. By establishing OMs for both the ODs and VOFDs of the GSJPs, we integrate them into the SCM, enabling efficient and accurate numerical computations. An error analysis and convergence study are conducted to validate the efficacy of the proposed algorithm. We demonstrate the applicability and accuracy of our method through eight numerical examples. Comparative analyses with prior research highlight the improved accuracy and efficiency achieved by our approach. The recommended approach exhibits excellent agreement between approximate and precise results in tables and graphs, demonstrating its high accuracy. This research contributes to the advancement of numerical methods for ODEs and MTVOFDEs with BCs, providing a reliable and efficient tool for solving complex BVPs with exceptional accuracy.

We introduce an efficient numerical method for second order linear ODEs whose solution may vary between highly oscillatory and slowly changing over the solution interval. In oscillatory regions the solution is generated via a nonoscillatory phase function that obeys the nonlinear Riccati equation. We propose a defect-correction iteration that gives an asymptotic series for such a phase function; this is numerically approximated on a Chebyshev grid with a small number of nodes. For analytic coefficients we prove that each iteration, up to a certain maximum number, reduces the residual by a factor of order of the local frequency. The algorithm adapts both the step size and the choice of method, switching to a conventional spectral collocation method away from oscillatory regions. In numerical experiments we find that our proposal outperforms other state-of-the-art oscillatory solvers, most significantly at low-to-intermediate frequencies and at low tolerances, where it may use up to 10 6 times fewer function evaluations. Even in high frequency regimes, our implementation is on average 10 times faster than other specialized solvers.

Acute circulatory failure (ACF) is a clinical syndrome when the heart and circulatory circulation cannot provide adequate blood supply to meet metabolic needs of the organs. ACF affects 30%- 50% of intensive care unit (ICU) patients. Fluid resuscitation is the primary treatment of ACF. However, it fails in a significant proportion (about 50%) of cases due to lack of clinically feasible non-invasive perfusion markers to assess the efficacy of the fluid therapy. Unfortunately, unsuccessful fluid therapy negatively affects patient outcome, increasing ICU length of stay and costs. Recent studies show identifying Stressed Blood Volume (SBV) of the cardiovascular system can be used to assess the potential efficacy of fluid therapy. The development of the diagnostic method requires the identification of the central arterial pressure curve based on the femoral arterial pressure, which is clinically available. This central arterial pressure curve can be used to identify the cardiovascular system parameters. In this study, the main goal was to develop a parameter-identification method for the Tube-load model-based transfer function connecting the femoral and central arterial pressure curve by using the so-called Physics-informed Neural Network methodology, namely the Neural ODE method. The study presents the adaptation of the Neural ODE method to the given parameter identification problem and the validation of the developed identification method. The robustness of the developed identification method was tested and used on a series of measurement data recorded in animal experiments.