Difference between revisions of "Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations"

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For a discrete-time example, let us again consider the Burger's equation but only allow ourselves data at two time snapshots. Specifically, our known data consists of 199 points at time <math display="inline"> t=0.1 </math>
 
For a discrete-time example, let us again consider the Burger's equation but only allow ourselves data at two time snapshots. Specifically, our known data consists of 199 points at time <math display="inline"> t=0.1 </math>
 
and 201 points at <math display="inline"> t=0.9 </math>. The correct parameter values and dataset noise is the same as in the continuous case and the procedure is as explained in the discrete-time section above. The neural network consists of four layers with 50 neurons per hidden layer. We choose the number of Runge-Kutta stages to be <math display="inline"> q=500 </math>, meaning that we approximate the solution at 500 intermediate time points. Note that the theoretical error estimates for a Runge-Kutta scheme with 500 stages is far below machine precision (truncation error of <math display="inline"> O(\Delta t^{2q}) </math>).
 
and 201 points at <math display="inline"> t=0.9 </math>. The correct parameter values and dataset noise is the same as in the continuous case and the procedure is as explained in the discrete-time section above. The neural network consists of four layers with 50 neurons per hidden layer. We choose the number of Runge-Kutta stages to be <math display="inline"> q=500 </math>, meaning that we approximate the solution at 500 intermediate time points. Note that the theoretical error estimates for a Runge-Kutta scheme with 500 stages is far below machine precision (truncation error of <math display="inline"> O(\Delta t^{2q}) </math>).
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The results of this example can be seen in figure 3

Revision as of 19:54, 13 November 2020

Presented by

Cameron Meaney

Introduction

In recent years, there has been an enormous growth in the amount of data and computing power available to researchers. Unfortunately, for many real-world scenarios, the cost of data acquisition is simply too high to collect an amount of data sufficient to guarantee robustness or convergence of training algorithms. In such situations, researchers are faced with the challenge of trying to generate results based on partial or incomplete datasets. Regularization techniques or methods which can artificially inflate the dataset become particularly useful in these situations; however, such techniques are often highly dependent of the specifics of the problem.

Luckily, in important real-world scenarios that we endeavor to analyze, there is often a wealth of existing information from which we can draw. This existing information commonly manifests in the form of a mathematical model, particularly a set of partial differential equations (PDEs). In this paper, the authors provide a technique for incorporating the information of a physical system contained in a PDE into the optimization of a deep neural network. This technique is most useful in situations where established PDE models exist, but where our amount of available data is too small for neural network training. In essence, the accompanying PDE model can be used as a regularization agent, constraining the space of acceptable solutions to help the optimization converge more quickly and more accurately.

Problem Setup

Consider the following general PDE

\begin{align*} u_t + N[u;\vec{\lambda}] = 0 \end{align*}

where the [math] u [/math] is the function we wish to find, subscripts denote partial derivatives, [math] \vec{\lambda} [/math] is the set of parameters on which the PDE depends, and [math] N [/math] is a differential, potentially nonlinear operator. This general form encompasses a wide array of PDEs used across the physical sciences including conservation laws, diffusion processes, advection-diffusion-reaction systems, and kinetic equations. Suppose that we have noisy measurements of the PDE solution, [math] u [/math] scattered across the spatio-temporal input domain. Then, we are interested in answering two questions about the physical system:

(1) Given fixed model parameters [math] \vec{\lambda} [/math], what can be said about the unknown hidden state [math] u(t,x) [/math]?

and

(2) What parameters [math] \vec{\lambda} [/math] best describe the observed data?

Data-Driven Solutions of PDEs

We will begin by attempting to answer the first of the questions above. Specifically, if given a small amount of noisy measurements of the solution of the PDE

\begin{align*} u_t + N[u] = 0, \end{align*}

can we estimate the full solution, [math] u(t,x) [/math], by approximating it with a deep neural network? Approximating the solution of the PDE with a neural network results in what the authors refer to as a 'Physics-Informed Neural Network' (PINN). Importantly, this technique is most useful when we are in the small-data regime - for if we had lots of data, it simply wouldn't be necessary to include information from the PDE because the data alone would be sufficient. In these examples, we are seeking to learn from a very small amount of data which makes information from the PDE necessary to include.


Continuous-Time Models

Consider the case where our noisy measurements of the solution are randomly scattered across the spatio-temporal input domain. This case is referred to as the 'continuous-time case.' Define the function

\begin{align*} f = u_t + N[u] \end{align*}

as the left hand side of the PDE above. Now assume that [math] u(t,x) [/math] can be approximated by a deep neural network. Therefore, the function [math] f(t,x) [/math] can also be approximate by a neural network since it is simply a function of [math] u(t,x) [/math]. In order to calculate [math] f(t,x) [/math] as a function of [math] u(t,x) [/math], derivates of [math] u(t,x) [/math] will need to be taken with respect to the inputs which is accomplished using a technique called automatic differentiation [?]. Importantly, the weights of the two neural networks will be shared, since [math] f(t,x) [/math] is simply a function of [math] u(t,x) [/math]. In order to find this set of weights, we create a loss function which has two distinct parts. The first part quantifies how well the neural network satisfies the known data points and is given by:

\begin{align*} MSE_u = \frac{1}{N} \sum_{i=1}^{N} [u(t_u^i,x_u^i) - u^i]^2 \end{align*}

where the summation is over the set of known data points. The second part of the loss function quantifies how well the neural network satisfies the PDE and is given by:

\begin{align*} MSE_f = \frac{1}{N} \sum_{i=1}^{N} [f(t_u^i,x_u^i)]^2. \end{align*}

The full loss function used in the optimization is then taken to be the sum of these two functions:

\begin{align*} MSE = MSE_u + MSE_f. \end{align*}

By using this loss function in the optimization, information from both the known data and the known physics (from PDE) can be incorporated into the neural network. This allows the network approximate the function by training on only a small number of data points. An example of this method can be seen in figure ?.


Discrete-Time Models

Now consider the case where our available data is not randomly scattered across the spatio-temporal domain, but rather only present at two particular times. This is known as the discrete-time case and occurs frequently in real-world examples such as when dealing with discrete pictures or medical images with no data between them. These cases and can be dealt with in the same manner as the continuous case with a few small adjustments. To adapt the PINN technique to discrete time models, we must leverage Runge-Kutta methods for numerical solutions of differential equations [?]. Runge-Kutta methods approximate the solution of a differential equation at the next numerical time step by first approximating the solution at a set of intermediate points between the time steps, then using these values to predict the full time step. The general form of a Runge-Kutta method with [math] q [/math] stages is given by:

\begin{align*} u^{n+c_i} &= u^n - \Delta t \sum^q_{j=1} a_{ij} N[u^{n+c_j}], ~ i = 1,...,q \\ u^{n+1} &= u^n - \Delta t \sum^q_{j=1} b_j N[u^{n+c_j}] \end{align*}

where [math] u^{n+c_j} = u(t^n + c_j \Delta t, x) [/math] and the general form includes both explicit and implicit time-stepping schemes. For more information of Runge-Kutta methods, see [?].

In the continuous-time case, we had approximated the function [math] u(t,x) [/math] by a neural network. Therefore, our neural network approximation for [math] u(t,x) [/math] had two inputs and one output. In the discrete case, instead of creating a neural netowrk which takes [math] t [/math] and [math] x [/math] as input and outputs the value of [math] u(t,x) [/math], we create a neural network which only takes [math] x [/math] and outputs the values of the solution at the intermediate points of the Runge-Kutta time-stepping scheme, [math] [u^{n+c_j}] [/math] for [math] i=1,...,q [/math]. Therefore, the PINN that we create here has one input and [math] q [/math] outputs. Importantly, information from the PDE is now incorporated into the the Runge-Kutta time-stepping scheme, so we do not need to add a term to the loss function to include it. Instead, our discrete-time loss function consists of two parts - one to quantify agreement with the data at the time of the initial data snapshot and one to quantify the agreement with data at the final data snapshot. For an example of this, see figure ?.

Data-Driven Discovery of PDEs

After having answered the first question, we can turn our focus to the second question. Specifically, if given a small amount of noisy measurements of the solution of the PDE

\begin{align*} u_t + N[u;\vec{\lambda}] = 0, \end{align*}

can we estimate the values of the parameters, [math] \vec{\lambda} [/math], that best describe the observed data? The difference between this case and the above is that we no longer know the values of the parameters [math] \vec{\lambda} [/math] appearing in the PDE. The procedure is, in essence, unchanged other than we now treat the PDE parameters as trainable parameters of the neural network. Since the discovery case outlined here is an extension of the solution case outlined above, the examples given below include unknown parameters and should therefore cover the full procedure.


Examples

Continuous-Time Example

For an example of this method in action, consider a problem involving Burger's equation, given by:

\begin{align*} &u_t + uu_x - (0.01/\pi)u_{xx} = 0, ~ x \in [-1,1], ~ t \in [0,1], \\ &u(0,x) = -\sin(\pi x), \\ &u(t, -1) = u(t,1) = 0. \end{align*}

Notably, Burger's equation is known as a challenging problem to solve because of the shock (discontinuity) that forms after sufficiently large time. However, using PINNs, this shockwave is easily handled.

So, assume that we are given noisy measurements of the solution of Burger's equation scattered across the spatio-temporal domain. Also assume that we do not know the values of the parameters in Burger's equation - we only know the equation form:

\begin{align*} &u_t + \lambda_1 uu_x - \lambda_2 u_{xx} = 0. \end{align*}

Additionally, we also assume that we are ignorant of the initial conditions and boundary conditions which generate the solution. Importantly, information form the initial and boundary conditions is contained in the known data points. We define the function [math] f(t,x) [/math] as:

\begin{align*} f = u_t + \lambda_1 uu_x - \lambda_2 u_{xx} \end{align*}

and assume that [math] u(t,x) [/math] is approximated by a deep neural network - hence creating a PINN. Then, the shared parameters of the neural networks for [math] u(t,x) [/math] and [math] f(t,x) [/math] as well as the parameters [math] \lambda_1 [/math] and [math] \lambda_2 [/math] are simultaneously learned by minimizing the combined loss function [math] MSE = MSE_u + MSE_f [/math] as defined above.

For this example, assume that we have 2000 datapoints across the entire spatio-temporal domain (representing a mere 2.0% of the known data). The correct values of the parameters which are used to generate the datapoints are [math] \lambda_1 = 1.0 [/math] and [math] \lambda_2 = 0.01/\pi [/math]. Also assume that the value of the solution for each of the known datapoints is randomly perturbed by up to 1% of its value - making the dataset noisy. This problem is trained using the procedure outlined above with a deep neural network of 9 layers with 20 neurons per hidden layer and using the L-BFGS optimizer.

The results of this example can be seen in figure 1. In the top panel, the exact solution can be seen with the datapoints selected for training pointed out. In the middle panel, a comparison of the exact and predicted solutions can be seen for three different times showing the accuracy of the PINN prediction. In the bottom panel, a comparison of the exact and predicted parameter values can also be seen. Also included in this bottom panel is the parameter predictions for the noiseless data case for comparison. Notice the remarkable accuracy with which the PINN is able to predict the correct parameter values in both noisy and noiseless cases. In figure 2, a comparison of the error in the predicted parameter values for different amounts of known data and noise.


Discrete-Time Example

For a discrete-time example, let us again consider the Burger's equation but only allow ourselves data at two time snapshots. Specifically, our known data consists of 199 points at time [math] t=0.1 [/math] and 201 points at [math] t=0.9 [/math]. The correct parameter values and dataset noise is the same as in the continuous case and the procedure is as explained in the discrete-time section above. The neural network consists of four layers with 50 neurons per hidden layer. We choose the number of Runge-Kutta stages to be [math] q=500 [/math], meaning that we approximate the solution at 500 intermediate time points. Note that the theoretical error estimates for a Runge-Kutta scheme with 500 stages is far below machine precision (truncation error of [math] O(\Delta t^{2q}) [/math]).

The results of this example can be seen in figure 3