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This paper compared physics-informed neural network (PINN), conventional neural network (NN) and traditional numerical discretization methods on solving differential equations (DEs) through literature investigation and experimental validation. We focused on the soft-constrained PINN approach and formalized its mathematical framework and computational flow for solving Ordinary DEs and Partial DEs (ODEs/PDEs). The working mechanism and its accuracy and efficiency were experimentally verified by solving typical linear and non-linear oscillator ODEs. We demonstrate that the DeepXDE-based implementation of PINN is not only light code and efficient in training, but also flexible across CPU/GPU platforms. PINN greatly reduces the need for labeled data: when the nonlinearity of the ODE is weak, a very small amount of supervised training data plus a few unsupervised collocation points are sufficient to predict the solution; in the minimalist case, only one or two training points (with initial values) are needed for first- or second-order ODEs, respectively. We also find that, with the aid of collocation points and the use of physical information, PINN has the ability to extrapolate data outside the time domain of the training set, and especially is robust to noisy data, thus with enhanced generalization capabilities. Training is accelerated when the gains obtained along with the reduction in the amount of data outweigh the delay caused by the increase in the loss function terms. The soft-constrained PINN can easily impose a physical law (e.g., conservation of energy) constraint by adding a regularization term to the total loss function, thus improving the solution performance to ODEs that obey this physical law. Furthermore, PINN can also be used for stiff ODEs, PDEs, and other types of DEs, and is becoming a favorable catalyst for the era of Digital Twins., Comment: 24 pages, 7 figures, 2 tables, etc. Ready for submission

A swarm algorithm framework (SWAF), realized by agent-based modeling, is presented to solve numerical optimization problems. Each agent is a bare bones cognitive architecture, which learns knowledge by appropriately deploying a set of simple rules in fast and frugal heuristics. Two essential categories of rules, the generate-and-test and the problem-formulation rules, are implemented, and both of the macro rules by simple combination and subsymbolic deploying of multiple rules among them are also studied. Experimental results on benchmark problems are presented, and performance comparison between SWAF and other existing algorithms indicates that it is efficiently., Comment: Genetic and Evolutionary Computation Conference (GECCO), Part I, 2004: 238-250 (LNCS 3102)

The periodic mode is analyzed together with two conventional boundary handling modes for particle swarm. By providing an infinite space that comprises periodic copies of original search space, it avoids possible disorganizing of particle swarm that is induced by the undesired mutations at the boundary. The results on benchmark functions show that particle swarm with periodic mode is capable of improving the search performance significantly, by compared with that of conventional modes and other algorithms., Comment: Congress on Evolutionary Computation, 2004. CEC2004. Volume: 2, On page(s): 2307- 2311 Vol.2

The adaptive constraints relaxing rule for swarm algorithms to handle with the problems with equality constraints is presented. The feasible space of such problems may be similiar to ridge function class, which is hard for applying swarm algorithms. To enter the solution space more easily, the relaxed quasi feasible space is introduced and shrinked adaptively. The experimental results on benchmark functions are compared with the performance of other algorithms, which show its efficiency., Comment: Congress on Evolutionary Computation, 2004. CEC2004. Volume: 2, On page(s): 2012- 2016 Vol.2

A self-organizing particle swarm is presented. It works in dissipative state by employing the small inertia weight, according to experimental analysis on a simplified model, which with fast convergence. Then by recognizing and replacing inactive particles according to the process deviation information of device parameters, the fluctuation is introduced so as to driving the irreversible evolution process with better fitness. The testing on benchmark functions and an application example for device optimization with designed fitness function indicates it improves the performance effectively., Comment: Congress on Evolutionary Computation, 2004. CEC2004. Volume: 2, On page(s): 2017- 2022 Vol.2

A dissipative particle swarm optimization is developed according to the self-organization of dissipative structure. The negative entropy is introduced to construct an opening dissipative system that is far-from-equilibrium so as to driving the irreversible evolution process with better fitness. The testing of two multimodal functions indicates it improves the performance effectively, Comment: Proceedings of the 2002 Congress on Evolutionary Computation, 2002. Volume: 2, On page(s): 1456-1461