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Solving Ordinary and Partial Differential Equations with Neural Networks(PINNs)
- An Ordinary Differential Equation (ODE) is a fundamental mathematical equation that relates a function to its derivatives.
- Partial Differential Equations (PDEs) have been critical in modeling physical phenomena in fields like thermodynamics, electrodynamics, wave dynamics, heat transfer, and many others.
- Computational methods aim to simulate intricate problems using PDEs and resolve them through numerical computation, developing accurate and efficient techniques to approximate solutions.
- Deep learning models can solve both ODEs and PDEs, converting them into the optimization problem by considering an independent function as an input to the neural network and the output as the dependent function variable of the given differential equation.
- The Physics-Informed Neural Network (PINN) method can be used for non-linear PDEs, such as Burgers' Equation, and provides data-driven solutions.
- In PINN, the problem is treated as a physical constraint problem with respect to neural networks from the physical world, where the loss function is determined based on backpropagation or forward propagation for minimizing the approximation functions.
- The Universal Approximation Theorem is a fundamental result in the field of ANN that states that certain types of neural networks can approximate specific functions to any desired degree of accuracy, enabling the network to learn complex patterns and relationships in data.
- The loss function quantifies how well or poorly a model is performing by calculating the difference between predicted and actual values, and guides the optimization process to enhance model accuracy.
- The backpropagation is a machine learning algorithm that trains neural networks by correcting errors, which is used to find the derivative of the function with respect to the input.
- The method of solving ODE and PDE using neural networks is a clever approach for simulating complex problems in the real world, which can not be addressed using traditional experimental or theoretical methods.
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