Reduced basis approximations of parameterized dynamical partial differential equations via neural networks

Nonlinear inequalities are widely used in science and engineering areas, attracting the attention of many researchers. In this article, a novel jump-gain integral recurrent (JGIR) neural network is proposed to solve noise-disturbed time-variant nonlinear inequality problems. To do so, an integral error function is first designed. Then, a neural dynamic method is adopted and the corresponding dynamic differential equation is obtained. Third, a jump gain is exploited and applied to the dynamic differential equation. Fourth, the derivatives of errors are substituted into the jump-gain dynamic differential equation, and the corresponding JGIR neural network is set up. Global convergence and robustness theorems are proposed and proved theoretically. Computer simulations verify that the proposed JGIR neural network can solve noise-disturbed time-variant nonlinear inequality problems effectively. Compared with some advanced methods, such as modified zeroing neural network (ZNN), noise-tolerant ZNN, and varying-parameter convergent-differential neural network, the proposed JGIR method has smaller computational errors, faster convergence speed, and no overshoot when disturbance exists. In addition, physical experiments on manipulator control have verified the effectiveness and superiority of the proposed JGIR neural network.

The objective of this research is to study the dynamic response characteristics of a three-beam system with intermediate elastic connections under a moving load/mass-spring. In this study, the finite Sine-Fourier transform was performed for the dynamic partial differential equations of a simply supported three-beam system (SSTBS) under a moving load and a moving mass-spring, respectively. The dynamic partial differential equations were transformed into dynamic ordinary differential equations relative to the time coordinates, and the equations were solved and the displacement Fourier amplitude spectral expressions were obtained. Finally, based on finite Sine-Fourier inverse transform, the expressions for dynamic response of SSTBS under the moving load and moving mass-spring were obtained. The proposed method, along with ANSYS, was used to calculate the dynamic response of the SSTBS under a moving load/mass-spring at different speeds. The results obtained herein were consistent with the ANSYS numerical calculation results, verifying the accuracy of the proposed method. The influence of the load/mass-spring’s moving speed on the dynamic deflections of SSTBS were analyzed. SSTBS has several critical speeds under a moving load/mass-spring. The vertical acceleration incurred by a change in the vertical speed of SSTBS due to the movement of mass-spring and the centrifugal acceleration produced by the movement of massspring on the vertical curve generated by SSTBS vibration could not be neglected.

Physics-informed neural networks for approximating dynamic (hyperbolic) PDEs of second order in time: Error analysis and algorithms

At present, the global energy problem is serious. Wave energy is the kinetic energy and potential energy of waves on the ocean surface. Because of its high energy density, wide distribution, and long continuous operation of the device, it has become the focus of academic research. The key problem to be solved urgently for the large-scale application of wave energy is how to optimize the energy conversion efficiency of wave energy devices. This paper starts from the Newtonian dynamic equation, analyzes the motion state of the wave energy device in different wave environments, and studies the optimization and control of its power output. This paper first adopts the analysis method of first whole and then isolation, and establishes a one-dimensional coordinate system with the zero point of sea level. Then, the force analysis of the system is carried out, the dynamic differential equation is established, and the initial draft of the system is solved through the initial balance equation. Since the dynamic equation is a second-order linear ordinary differential inhomogeneous linear equation, the fourth-order Runge is used. The Kutta method is used to solve the problem, and the numerical solution for the position and velocity of the float is obtained. Secondly, the movable coordinate system is established with the float as the reference system, and the force analysis of the oscillator is carried out to obtain its relative dynamic equation. Then, the initial relative position of the oscillator is solved through the initial balance equation, and the relative dynamic differential equation is obtained by using the fourth-order Runge-Kutta method to obtain the numerical solution of the relative position and relative velocity of the oscillator with respect to time. Finally, the numerical solution of the absolute position and absolute velocity of the oscillator with respect to time is obtained through coordinate transformation.

Simulation with position-based dynamics is very popular due to its high efficiency. However, it has the weaknesses of lacking details when too few vertices are involved in simulation and inefficiency when too many vertices are used for simulation. To tackle this problem, in this paper, we propose a new method of reconstructing dynamic 3D models with small data. The core elements of the proposed approach are a curve-represented geometric model and a physics-based mathematical model defined by dynamic partial differential equations. We first use the simulation method of position-based dynamics to generate a group of keyframe poses, which are used to create the deformation animation of a 3D model. Then, wireframe curves are extracted from skin deformation shapes of the 3D model at different keyframe poses. A physics-based mathematical model defined by dynamic partial differential equations is proposed. Its closed-form solution is obtained to represent the extracted curves, which are used to reconstruct the deformation models at different keyframe poses. Experimental examples and comparisons made in this paper indicate that the proposed method of reconstructing dynamic 3D models can greatly reduce data size while keeping good details.

In this contribution, a novel hybrid approach involving artificial neural networks (ANNs) and heuristic algorithms is employed for Hall currents and electromagnetic effects analysis for a multi-phase wavy flow. The governing partial differential equations (PDEs) for flow dynamics are reduced into a corresponding system of ordinary differential equations (ODEs) with a pertinent transformation technique. The novelty of this work is combination of Morlet wavelet and hyperbolic tangent (Tanh) functions is employed as an activation function in artificial neural networks (ANNs) to effectively capture nonlinear behavior for flow dynamics. The novel effective Morlet wavele Tanh neural networks (MTNNs) based fitness function is formulated for solution estimation of the model. The weights and biases of MTNNs are optimized with a global searching technique by particle swarm optimization (PSO). Numerical solution of ODEs is also obtained through Python physics informed neural networks (PINNs) with Adam optimizer for validation of the proposed solutions. Statistical analysis involving histogram visualizations, probability plots, and boxplots is performed for accuracy, robustness, convergence, and stability evaluation of the proposed solution with respect to crucial error measures such as cost function, absolute error, and mean squared error (MSE). The MSE values for velocity and temperature range from [Formula: see text] to [Formula: see text] and [Formula: see text] to [Formula: see text], respectively. Graphical analysis reveals that flow velocity and thermal distributions are influenced directly by electroosmotic factor but are affected inversely with the applied magnetic field. The proposed MTNNs yield results that closely align with those obtained using PINNs.

Scientific machine learning is a burgeoning discipline which blends scientific computing and machine learning.Traditionally, scientific computing focuses on large-scale mechanistic models, usually differential equations, that are derived from scientific laws that simplified and explained phenomena.On the other hand, machine learning focuses on developing non-mechanistic data-driven models which require minimal knowledge and prior assumptions.The two sides have their pros and cons: differential equation models are great at extrapolating, the terms are explainable, and they can be fit with small data and few parameters.Machine learning models on the other hand require "big data" and lots of parameters but are not biased by the scientists ability to correctly identify valid laws and assumptions.However, the recent trend has been to merge the two disciplines, allowing explainable models that are data-driven, require less data than traditional machine learning, and utilize the knowledge encapsulated in centuries of scientific literature.The promise is to fuse a priori domain knowledge which doesn't fit into a "dataset", allow this knowledge to specify a general structure that prevents overfitting, reduces the number of parameters, and promotes extrapolatability, while still utilizing machine learning techniques to learn specific unknown terms in the model.This has started to be used for outcomes like automated hypothesis generation and accelerated scientific simulation.The purpose of this blog post is to introduce the reader to the tools of scientific machine learning, identify how they come together, and showcase the existing open source tools which can help one get started.We will be focusing on differentiable programming frameworks in the major languages for scientific machine learning: C++, Fortran, Julia, MATLAB, Python, and R.We will be comparing two important aspects: efficiency and composability.Efficiency will be taken in the context of scientific machine learning: by now most tools are well-optimized for the giant neural networks found in traditional machine learning, but, as will be discussed here, that does not necessarily make them efficient when deployed inside of differential equation solvers or when mixed with probabilistic programming tools.Additionally, composability is a key aspect of scientific machine learning since our toolkit is not ML in isolation.Our goal is not to do machine learning as seen in a machine learning conference (classification, NLP, etc.), and it's not to do traditional machine learning as applied to scientific data.Instead, we are putting ML models and techniques into the heart of scientific simulation tools to accelerate and enhance them.Our neural networks need to fully integrate with tools that simulate satellites and robotics simulators.They need to integrate with the packages that we use in our scientific work for verifying numerical accuracy, tracking units, estimating uncertainty, and much more.We need our neural networks to play nicely with existing packages for delay

Jacobi, in a posthumous memoir* which has only this year appeared, has developed two remarkable methods (agreeing in their general character, but differing in details) of solving non-linear partial differential equations of the first order, and has applied them in connexion with that theory of the Differential Equations of Dynamics which was esta­blished by Sir W. B. Hamilton in the Philosophical Transactions for 1834‒35. The knowledge, indeed, that the solution of the equations of a dynamical problem is involved in the discovery of a single central function, defined by a single partial differential equation of the first order, does not appear to have been hitherto (perhaps it will never be) very fruitful in practical results. But in the order of those speculative truths which enable us to perceive unity where it was unperceived before, its place is a high and enduring one. Given a system of dynamical equations, it is possible, as Jacobi had shown, to con­struct a partial differential equation such that from any complete primitive of that equation, i. e. from any solution of it involving a number of constants equal to the number of the independent variables, all the integrals of the dynamical equations can be deduced by processes of differentiation. Hitherto, however, the discovery of the com­plete primitive of a partial differential equation has been supposed to require a previous knowledge of the integrals of a certain auxiliary system of ordinary differential equa­tions; and in the case under consideration that auxiliary system consisted of the dynamical equations themselves. Jacobi’s new methods do not require the preliminary integration of the auxiliary system. They require, instead of this, the solution of certain systems of simultaneous linear partial differential equations. To this object therefore the method developed in my recent paper on Simultaneous Differential Equa­tions might be applied. But the systems of equations in question are of a peculiar form. They admit, in consequence of this, of a peculiar analysis. And Jacobi’s methods of solving them are in fact different from the one given by me, though connected with it by remarkable relations. He does indeed refer to the general problem of the solution of simultaneous partial differential equations, and this in language which does not even suppose the condition of linearity. He says, “Non ego hic immorabor qusestioni generali quando et quomodo duabus compluribusve æquationibus differentialibus partialibus una eademque functione Satisfied possit, sed ad casum propositum investigationem restringam. Quippe quo præclaris uti licet artificiis ad integrationem expediendam commodis. ” But he does not, as far as I have been able to discover, discuss any systems of equations more general than those which arise in the immediate problem before him.

Numerical weather prediction revisions using the locally trained differential polynomial network

Growth of solutions for measure differential equations and dynamic equations on time scales

A simulation method of an intelligent contactor is presented by using a neural network to fit the proven relationship among the flux linkage, the electrical current, and the moving core displacement of a contactor in this article. First, the neural network algorithm is trained by the operational data of a contactor driven by a basic training circuit to solve the coil current. Then, a dynamic simulation program of the contactor model is constructed via combining the algorithm and dynamic differential equations. On this basis, by means of the co-simulation technology, the point-by-point closed-loop simulation between the control module and the contactor model is carried out. Accordingly, the co-simulation of an intelligent contactor based on a neural network is completed. The simulation method can avoid the complex finite-element modeling of a contactor and realize the model extraction of an arbitrary contactor. The extracted model can be combined with a drive circuit and any control strategy to perform the co-simulation, which is convenient for the flexible design of hardware control circuits and software control strategies of various intelligent contactors.

A projection neural network method for circular cone programming is proposed. In the KKT condition for the circular cone programming, the complementary slack equation is transformed into an equivalent projection equation. The energy function is constructed by the distance function and the dynamic differential equation is given by the descent direction of the energy function. Since the projection on the circular cone is simple and costs less computation time, the proposed neural network requires less state variables and leads to low complexity. We prove that the proposed neural network is stable in the sense of Lyapunov and globally convergent. The simulation experiments show our method is efficient and effective.

The basic characteristics of mechanism kinematics are investigated to explain the background for the dynamic simulation procedure developed herein, which is especially suitable for pantographic deployable masts (PDM). Natural Cartesian coordinates are used due to their simplicity. The kinetic energy of the uniform rod can thus be expressed in terms of the Cartesian coordinates of the joints. First, dynamic differential equations are formulated with respect to a series of dependent coordinates. The constraint equations of the rigid body, scissor-like-element (SLE), and slider, etc., are formulated in succession. Then, based on the first and second differentials of the constraint equations and the null-space base of the constraint Jacobian matrix, we introduce the constraints into the differential dynamic equations, which are so reduced with respect to the independent generalized coordinates. Furthermore, a Runge–Kutta 4th-order (RK-4) algorithm, together with numerical stabilization techniques, is used to get the time history solutions of the dynamic equations. Finally, three numerical tests are presented, and dynamic deployment simulations are carried out for four particular PDM. The results show that the algorithm is efficient for dynamic simulations of a foldable truss. *Communicated by E. Zahariev.

Dynamic simulation is widely applied in the real-time and realistic physical simulation field. How to achieve natural dynamic simulation results in real-time with small data sizes is an important and long-standing topic. In this paper, we propose a dynamic reconstruction and interpolation method grounded in physical principles for simulating dynamic deformations. This method replaces the deformation forces of the widely used eXtended Position-Based Dynamics (XPBD), which are traditionally derived from the gradient of the energy potential defined by the constraint function, with the elastic beam bending forces to more accurately represent the underlying deformation physics. By doing so, it establishes a mathematical model based on dynamic partial differential equations (PDE) for reconstruction, which are the differential equations involving both the parametric variable u and the time variable t. This model also considers the inertia forces caused by acceleration. The analytical solution to this model is then integrated with the XPBD framework, built upon Newton’s equations of motion. This integration reduces the number of design variables and data sizes, enhances simulation efficiency, achieves good reconstruction accuracy, and makes deformation simulation more capable. The experiment carried out in this paper demonstrates that deformed shapes at about half of the keyframes simulated by XPBD can be reconstructed by the proposed PDE-based dynamic reconstruction algorithm quickly and accurately with a compact and analytical representation, which outperforms static B-spline-based representation and interpolation, greatly shortens the XPBD simulation time, and represents deformed shapes with much smaller data sizes while maintaining good accuracy. Furthermore, the proposed PDE-based dynamic reconstruction algorithm can generate continuous deformation shapes, which cannot be generated by XPBD, to raise the capacity of deformation simulation.

Dynamic PDE parametric curves