Semi-empirical Neural Network Modeling presents a new approach on how to quickly construct an accurate, multilayered neural network solution of differential equations. Current neural network methods have significant disadvantages, including a lengthy learning process and single-layered neural networks built on the finite element method (FEM). The strength of the new method presented in this book is the automatic inclusion of task parameters in the final solution formula, which eliminates the need for repeated problem-solving. This is especially important for constructing individual models with unique features. The book illustrates key concepts through a large number of specific problems, both hypothetical models and practical interest.
1. Introduction - p. xv 2. 1: Examples of problem statements and functionals - p. xlvii 3. Abstract - p. 1 4. 1.1 Problems for ordinary differential equations - p. 4 5. 1.2 Problems for partial differential equations for domains with fixed boundaries - p. 13 6. 1.3 Problems for partial differential equations in the case of the domain with variable borders - p. 28 7. 1.4 Inverse and other ill-posed problems - p. 36 8. 2: The choice of the functional basis (set of bases) - p. 49 9. 3: Methods for the selection of parameters and structure of the neural network model - p. 72 10. 4: Results of computational experiments - p. 103 11. Abstract - p. 105 12. 4.1 Solving problems for ordinary differential equations - p. 105 13. 4.2 Solving problems for partial differential equations in domains with constant boundaries - p. 126 14. 4.3 Solving problems for partial differential equations for domains with variable boundaries - p. 149 15. 4.4 Solving inverse and other ill-posed problems - p. 151 16. 5: Methods for constructing multilayer semi-empirical models - p. 171 17. Abstract - p. 173 18. 5.1 General description of methods - p. 173 19. 5.2 Application of methods for constructing approximate analytical solutions for ordinary differential equations - p. 181 20. 5.3 Application of multilayer methods for partial differential equations - p. 208 21. 5.4 Problems with real measurements - p. 215 22. Index - p. 237
Semi-empirical Neural Network Modeling presents a new approach on how to quickly construct an accurate, multilayered neural network solution of differential equations. Current neural network methods have significant disadvantages, including a lengthy learning process and single-layered neural networks built on the finite element method (FEM). The strength of the new method presented in this book is the automatic inclusion of task parameters in the final solution formula, which eliminates the need for repeated problem-solving. This is especially important for constructing individual models with unique features. The book illustrates key concepts through a large number of specific problems, both hypothetical models and practical interest.
1. Introduction - p. xv 2. 1: Examples of problem statements and functionals - p. xlvii 3. Abstract - p. 1 4. 1.1 Problems for ordinary differential equations - p. 4 5. 1.2 Problems for partial differential equations for domains with fixed boundaries - p. 13 6. 1.3 Problems for partial differential equations in the case of the domain with variable borders - p. 28 7. 1.4 Inverse and other ill-posed problems - p. 36 8. 2: The choice of the functional basis (set of bases) - p. 49 9. 3: Methods for the selection of parameters and structure of the neural network model - p. 72 10. 4: Results of computational experiments - p. 103 11. Abstract - p. 105 12. 4.1 Solving problems for ordinary differential equations - p. 105 13. 4.2 Solving problems for partial differential equations in domains with constant boundaries - p. 126 14. 4.3 Solving problems for partial differential equations for domains with variable boundaries - p. 149 15. 4.4 Solving inverse and other ill-posed problems - p. 151 16. 5: Methods for constructing multilayer semi-empirical models - p. 171 17. Abstract - p. 173 18. 5.1 General description of methods - p. 173 19. 5.2 Application of methods for constructing approximate analytical solutions for ordinary differential equations - p. 181 20. 5.3 Application of multilayer methods for partial differential equations - p. 208 21. 5.4 Problems with real measurements - p. 215 22. Index - p. 237
