Using the concept of Physics informed Neural Networks(PINNs) derived from the Cited Reference Paper we solve a 1D Heat Transfer equation. Problem Statement:
- We have been given a PDE: du/dx=2du/dt+u and boundary condition: u(x,0)=10e^(-5x)
- Independent variables: x,t (input) Dependent variables: u (outputs) We have to find out u(x,t) for all x in range [0,10] and t in range [0,10]
- Our function is f = du/dx - 2du/dt - u
We exploit recent developments in automatic differentiation – one of the most useful but perhaps under-utilized techniques in scientific computing – to differentiate neural networks with respect to their input coordinates and model parameters to obtain physics-informed neural networks.
In this work, we consider parametrized and nonlinear partial differential equations of the general form: ut + N [u; λ] = 0, x ∈ , t ∈ [0, T ], where u(t, x) denotes the latent (hidden) solution, N [·; λ] is a nonlinear operator parametrized by λ, and is a subset of RD . This setup encapsulates a wide range of problems in mathematical physics including conservation laws, diffusion processes, advection–diffusion–reaction systems, and kinetic equations
Citing Research Paper: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
M. Raissi, P. Perdikaris, G.E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational Physics, Volume 378, 2019, Pages 686-707, ISSN 0021-9991, https://doi.org/10.1016/j.jcp.2018.10.045. (https://www.sciencedirect.com/science/article/pii/S0021999118307125) Abstract: We introduce physics-informed neural networks – neural networks that are trained to solve supervised learning tasks while respecting any given laws of physics described by general nonlinear partial differential equations. In this work, we present our developments in the context of solving two main classes of problems: data-driven solution and data-driven discovery of partial differential equations. Depending on the nature and arrangement of the available data, we devise two distinct types of algorithms, namely continuous time and discrete time models. The first type of models forms a new family of data-efficient spatio-temporal function approximators, while the latter type allows the use of arbitrarily accurate implicit Runge–Kutta time stepping schemes with unlimited number of stages. The effectiveness of the proposed framework is demonstrated through a collection of classical problems in fluids, quantum mechanics, reaction–diffusion systems, and the propagation of nonlinear shallow-water waves. Keywords: Data-driven scientific computing; Machine learning; Predictive modeling; Runge–Kutta methods; Nonlinear dynamics
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