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Solving differential equations is still a topic of major interest, due to their appearance in many fields of science and engineering and a classic approach with neural networks builds upon trial solutions, the so-called neural forms. The latter are incorporated in a cost function that is subject to minimisation, to train the involved neural networks. Neural forms represent general and flexible tools for solving ordinary differential equations, partial differential equations as well as systems of each. However, the computational approach is in general highly dependent on a variety of computational parameters and the choice of the optimisation methods. Studying the solution of a simple but fundamental stiff ordinary differential equations with small feedforward neural networks and first order optimisation shows, that it is possible to identify preferable choices for parameters and methods. The neural network weight initialisation appears to be a sensitive topic, while having a major impact on the solution accuracy. Especially the use of non-random (deterministic) weights partially shows poor performance, but removes a stochastic component. Further research reveals, that a new polynomial representation of the neural forms can significantly increase the reliability of a deterministic initialisation (all weights have initially the same values assigned). In order to maintain smaller neural network architectures and solve the differential equation, even on fairly large domains, a new technique called domain segmentation (for initial value problems) is introduced. The solution domain splits into equidistant subdomains and the above-mentioned collocation polynomial neural forms are solved separately in each domain fragment. At the boundary of any subdomain, a new initial value is provided by the neural forms solution and directly incorporated in the adjacent one. In classic adaptive numerical methods for solving differential equations, the mesh as well as the domain may be refined or decomposed, respectively, in order to improve numerical accuracy. The subdomain distribution can also be connected with an adaptive refinement. That is, the neural network training status is combined with an adaptive subdomain size reduction in the new adaptive neural domain refinement algorithm. That is, each subdomain is reduced in size until the optimisation is resolved up to a predefined training accuracy. In addition, while the neural networks are by default small, the number of neurons may also be adjusted in an adaptive way. Conditions are introduced to automatically confirm the solution reliability and optimise computational parameters whenever it is necessary.
In this work Adaptive Polynomial Tabulation (APT) is presented. It is a new approach to solve the initial value chemical rate equation system. In this approach zeroeth, first and second order polynomials are used in real-time to approximate the solution of the initial value chemical rate equation system. The sizes of the local regions encountered for the different orders of polynomial approximation are calculated in real-time. To improve accuracy the chemical state space is partitioned into hypercubes. During calculations the hypercubes accessed by the reactive mixture are divided into adaptive hypercubes depending on the accuracy of the local solution. Mixture initial conditions are stored in the adaptive hypercubes. Around each stored initial condition two concentric ellipsoids of accuracy (EOA) are defined. These include the ISAT and identical EOAs. The time evolution of mixture initial conditions which encounter an identical and ISAT EOA are approximated by zero and first order polynomials respectively. With a certain number of stored initial conditions within an adaptive hypercube, its second order polynomial coefficients are constructed from the stored initial conditions. The time evolution of additional mixture initial conditions that encounter this adaptive hypercube are approximated with second order polynomials. The APT model is simplified by the replacement of the entire set of species mass fractions with a progress variable based on the enthalpy of formation evaluated at 298 K. APT has 3 degrees of freedom which include the progress variable, total enthalpy and pressure. The APT model was tested with a zero dimensional Stochastic Reactor Model (SRM) for HCCI engine combustion. A skeletal n-heptane/toluene mechanism with 148 chemical species and 1281 reactions was used. In the tests, the HCCI engine simulations using APT were in very good agreement with the model calculations using the ODE solver. The cool flame and main ignition events were accurately captured. The major and minor species were also accurately captured by APT. In SRM-HCCI calculations without cyclic variations, a computational speed up factor greater than 1000 was obtained when APT was used for all the operating points considered without significant loss in accuracy. For the SRM-HCCI engine calculations with cyclic variations, APT demonstrated a computational speed up exceeding 12 without significant loss in accuracy.