In this thesis, we address the issue of boundary influence by interior points in the context of the non-local singularly perturbed equation. On the other hand, we provide an example with exact solutions and employ neural networks to analyze its solutions. Moreover, we identify several potential challenges that may arise. In the first section, we explore computational approaches to estimate the behavior of the solution and its derivatives. In the second section, we investigate the problems that may occur when using neural networks to handle such equations, along with their underlying causes.

1 Preliminary 6
2 Exploring the Solution of Singular Differential Equation with
Non-local Boundary Conditions by Calculating 13
2.1 Primary Idea about Observing The Behavior of v(x) And w(x)
Near Boundary Points . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Observing The Behavior of v(x) And w(x) in The Interior Domain 17
2.3 Observing The Behavior of v(x) And w(x) Near Boundary Points 24
2.4 Approximation of Boundary Condition u(0) And u(1) . . . . . . 27
2.5 Approximation of Derivative of Solution u(x) . . . . . . . . . . 31
3 Deep Learning Method 34
3.1 How to Approximate The Solution And What Methods We Give 35
3.1.1 Solving Ordinary Differential Equation by Artificial Neural
Network . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.2 Implementation Process for Two Methods . . . . . . . . 36
3.2 Addressing Challenges: Tips and Techniques . . . . . . . . . . . 38
3.2.1 Unstable Solutions Caused by Ill-Conditioning . . . . . . 38
3.2.2 A Challenging Problem Coming From Non-Local Boundary
Conditions . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.3 Optimization Process with Annealing Scheme . . . . . . 41
3.3 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Summary 48

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