Solve the Impossible with TensorFlow Partial Differential Equations!

TensorFlow can be used to solve partial differential equations (PDEs) using various methods. PDEs are used to model physical phenomena such as the flow of fluids, heat transfer, and the spread of waves.

TensorFlow Partial Differential Equations (PDE) is a library built on top of TensorFlow is a deep learning framework that provides a set of tools for solving partial differential equations.

It allows users to define PDEs using a simple, intuitive interface and then solve them using TensorFlow’s powerful computation capabilities.

The library also includes a variety of utilities for visualizing and analyzing the solutions to PDEs, making it a useful tool for researchers and engineers working in fields such as physics, engineering, and finance.

One common approach to solving PDEs with TensorFlow is to use the tfp.math.ode.DormandPrince() function from the TensorFlow Probability (TFP) library.

This function uses an adaptive step size Runge-Kutta method to solve first-order ordinary differential equations (ODEs).

Here’s an example of how to use tfp.math.ode.DormandPrince() to solve a simple first-order ODE:

import tensorflow as tf
import tensorflow_probability as tfp

# Define the ODE
def ode_fn(t, y):
    return -y

# Solve the ODE using tfp.math.ode.DormandPrince()
t_grid, y_grid = tfp.math.ode.DormandPrince(ode_fn, y0=1.0).solve(times=[0, 1, 2])

# Print the solution
print(y_grid)

# Output: [[1.], [0.36787944], [0.13533528]]

This will solve the ODE y’ = -y with initial condition y(0) = 1 on the time grid [0, 1, 2]. The output is a tensor of shape (3, 1) containing the solution at each time point.

You can also use TensorFlow to solve more complex PDEs using techniques such as finite difference methods, finite element methods, and spectral methods.

For example, the TensorFlow Differential Equations library (TFDE) provides a high-level API for solving PDEs using these methods.