| Titre : | Machine Learning methods for solving Stochastic Differential Equations |
| Auteurs : | HASSANI Maroua, Auteur ; MEKKAOUI Imane, Directeur de thèse |
| Type de document : | texte manuscrit |
| Editeur : | University of Saida - Dr Moulay Tahar. Faculty: Mathematics, Computer Science and Telecommunications Department: Mathematics, 2025/2026 |
| Format : | 87 ص |
| Accompagnement : | CD |
| Langues: | Anglais |
| Index. décimale : | BUC-M 008496 |
| Catégories : |
Master Mathématiques Spécialité: Analyse stochastique, statistique des processus et applications |
| Résumé : |
Stochastic differential equations (SDE) play an important role in modeling random
phenomena in many scientific fields, including finance, physics, and engineering. Classical numerical techniques, including Euler−Maruyama, or Monte Carlo are commonly used to estimate their responses. However, these methods may face limitations in terms of efficiency and scalability when dealing with complex or high-dimensional problems. This work deals with machine learning methods, especially Deep Learning methods for solving stochastic differential equations and compares them with classical numerical methods. We focus on commonly used methods in the literature such as Physics Informed Neural Networks (PINNs) and Deep Forward-Backward Stochastic Neural Networks. The results show that these techniques are promising in terms of providing accurate approximations and being also characterized by their ability to scale and efficiency in computations, especially in the context of high dimensions for nonlinear equations |
| Note de contenu : |
Contents
List of Figures 1 Introduction 4 1 Reminders about Stochastic Differential Equations 8 1.1 Stochastic differential equations . . . . . . . . . . . . . . . . . . . . . 8 1.2 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Existence and uniqueness of solutions . . . . . . . . . . . . . . . . . . 9 1.4 Classical Examples of SDEs . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Numerical Methods for SDEs . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Limitations of Classical Methods . . . . . . . . . . . . . . . . . . . . 14 2 Machine Learning and Deep Learning basics 16 2.1 Machine Learning basics . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.1 The main types of Machine Learning models . . . . . . . . . . 17 2.1.2 Notion of training algorithms in Machine Learning . . . . . . 17 2.1.3 Types of Machine Learning Algorithms . . . . . . . . . . . . . 18 2.2 Deep Learning principles . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.1 Types of deep learning models . . . . . . . . . . . . . . . . . . 20 2.2.2 Hyperparameters in Deep Learning . . . . . . . . . . . . . . . 21 2.2.3 Deep Learning architecture . . . . . . . . . . . . . . . . . . . . 22 2.2.3.1 Activation Functions in Neural Networks . . . . . . . 22 2.2.3.2 Multilayer Perceptron (MLP) . . . . . . . . . . . . . 23 2.2.3.3 Recurrent Neural Networks (RNNs) . . . . . . . . . . 24 2.2.3.4 Long Short-Term Memory (LSTM) . . . . . . . . . . 25 2.2.4 Loss function and back-propagation in neural networks . . . . 27 2.2.5 Challenges in Deep Learning . . . . . . . . . . . . . . . . . . . 30 2.3 Differences Between Machine Learning and Deep Learning . . . . . . 31 2.4 When to Use Machine Learning vs. Deep Learning . . . . . . . . . . 32 2.5 Advantages and limitations of Machine Learning and Deep Learning . 33 2 CONTENTS 3 Deep Learning-based PINNs method 35 3.1 Principle and motivation . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 The Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . . . . 36 3.3 How PINNs work for the Fokker-Planck Equation . . . . . . . . . . . 37 3.3.1 Step 1: Neural Network Approximation . . . . . . . . . . . . . 37 3.3.2 Step 2: Automatic Differentiation . . . . . . . . . . . . . . . . 39 3.3.3 Step 3: Residual of the PDE . . . . . . . . . . . . . . . . . . . 39 3.3.4 Step 4: Training Objective . . . . . . . . . . . . . . . . . . . . 40 3.4 The loss function in PINNs and Adam optimizer . . . . . . . . . . . . 40 3.4.1 Loss Function in PINNs . . . . . . . . . . . . . . . . . . . . . 40 3.4.2 Adam Optimizer in PINNs . . . . . . . . . . . . . . . . . . . . 41 3.5 Numerical experiments and comparative study . . . . . . . . . . . . . 42 3.6 Advantages and limitations of PINNs . . . . . . . . . . . . . . . . . . 59 3.6.1 Advantages: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.6.2 Limitations: . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4 Deep Learning-based FBSNNs methods 61 4.1 The Kolmogorov PDE and Forward-Backwad SDE . . . . . . . . . . . 62 4.2 How Deep FBSNNs works for the Kolmogorov equation . . . . . . . . 62 4.2.1 Step 1: Reformulate the PDE as an FBSDE System . . . . . . 63 4.2.2 Step 2: Time Discretization of the FBSDE . . . . . . . . . . . 63 4.2.3 Step 3: Neural Network Approximation and loss function . . . 64 4.3 Numerical simulations and comparative study . . . . . . . . . . . . . 65 4.4 Advantages and limitations of the FBSNNs method . . . . . . . . . . 75 4.4.1 Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.4.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Conclusion 78 References 80 3 |
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