Machine learning applied to proton radiography of high-energy-density plasmas

Proton radiography is a technique extensively used to resolve magnetic field structures in high-energy-density plasmas, revealing a whole variety of interesting phenomena such as magnetic reconnection and collisionless shocks found in astrophysical systems. Existing methods of analyzing proton radiographs give mostly qualitative results or specific quantitative parameters, such as magnetic field strength, and recent work showed that the line-integrated transverse magnetic field can be reconstructed in specific regimes where many simplifying assumptions were needed. Using artificial neural networks, we demonstrate for the first time 3D reconstruction of magnetic fields in the nonlinear regime, an improvement over existing methods, which reconstruct only in 2D and in the linear regime. A proof of concept is presented here, with mean reconstruction errors of less than 5% even after introducing noise. We demonstrate that over the long term, this approach is more computationally efficient compared to other techniques. We also highlight the need for proton tomography because (i) certain field structures cannot be reconstructed from a single radiograph and (ii) errors can be further reduced when reconstruction is performed on radiographs generated by proton beams fired in different directions.

Figure 1

Sample of an experimental radiograph. The whiter regions indicate areas with higher proton intensity.

Figure 2

Diagram of a typical proton radiography setup. A point source of distance $l$ away from the object emits a beam of protons moving generally in the $z$ direction. $L$ is the distance between the object and image plane.

Figure 3

A neuron, the basic unit in an artificial neural network. The input vector $\mathbf{x}$ is mapped to a scalar $y$ via a nonlinear function $S$. The connections represent the inputting of elements of $\mathbf{x}$ into the neuron, and each connection is assigned a weight, which is used in calculating the output.

Figure 4

Schematic of a feedforward neural network, where the outputs of one layer are fed into the inputs of an adjacent layer. It takes a vector $\mathbf{x}\in {\mathbb{R}}^n$ and outputs a vector $\mathbf{y}(\mathbf{x},\theta )\in {\mathbb{R}}^m$, where $\theta$ represents the parameters (weights and biases) of the neural network.

Figure 5

Schematic of the prediction and reconstruction process.

Figure 6

Typical curves of training and validation errors with respect to training iteration. Beyond a certain point, the artificial neural network starts to model noise, causing the validation error to increase. Training should be halted when this happens.

Figure 7

Error histograms for the $\alpha$ coefficients of two basis fields, using an artificial neural network with 1 hidden layer consisting of 10 neurons. The mean errors are $0.34\%$ and $2.74\%$, while the median errors are $0.20\%$ and $1.29\%$ for ${\alpha }_1$ (ellipsoidal blob, left) and ${\alpha }_2$ (flux rope, right), respectively. More parameters can be found in Table 1.

Figure 8

Left: Radiograph for a magnetic ellipsoidal blob at $B=0.1$ T, $\sigma$ = 1. This is in the noncaustic regime, where the ring around the center is smeared out. Right: Radiograph for a magnetic ellipsoidal blob at $B=0.3$ T, $\sigma$ = 1. This is in the caustic regime, where most of the protons fall into a very thin ring. The scales are in arbitrary units.

Figure 9

Error histograms for $\alpha$ (left) and $\sigma$ (right) for a magnetic ellipsoidal blob, using an artificial neural network with 1 hidden layer consisting of 50 neurons. The mean errors are $0.26\%$ and $0.05\%$ while the median errors are $0.20\%$ and $0.04\%$ for $\alpha$ and $\sigma$, respectively. More parameters can be found in Table 1.

Figure 10

Horizontal profile of a radiograph for a magnetic ellipsoidal blob at 1.5 T, the branching caustics regime. Notice that there are two maxima in the intensity profile, instead of one in the case of the caustic regime.

Figure 11

Error histogram of $\alpha$ for a magnetic ellipsoidal blob spanning the linear, caustic, and branching caustic regime, using an artificial neural network with 1 hidden layer consisting of 10 neurons. The mean error is $0.06\%$ while the median error is $0.04\%$. More parameters can be found in Table 1.

Figure 12

Error histograms of $\alpha$ and $\sigma$ when 5, 10, 20, and 30 percent noise is introduced into the radiographs for a magnetic ellipsoidal blob. The artificial neural network configurations are: 8 hidden layers with 80 neurons per layer, 5 hidden layers with 50 neurons per layer, 6 hidden layers with 70 neurons per layer, and 7 hidden layers with 100 neurons per layer for 5, 10, 20, and 30 percent noise, respectively. For $\alpha$, the mean errors are $0.92\%$, $1.14\%$, $1.49\%$, and $1.91\%$, while the median errors are $0.73\%$, $0.82\%$, $1.19\%$, and $1.46\%$ for 5, 10, 20, and 30 percent noise, respectively. For $\sigma$, the mean errors are $0.18\%$, $0.24\%$, $0.31\%$, and $0.41\%$, while the median errors are $0.14\%$, $0.18\%$, $0.25\%$, and $0.31\%$ for 5, 10, 20, and 30 percent noise, respectively. Notice that it takes an increase from $5\%$ noise to $30\%$ noise in order to roughly double the mean and median errors, indicating that the artificial neural network method is robust to noise. More parameters can be found in Table 1.

Figure 13

Error histogram in $\alpha$ for a magnetic ellipsoidal blob (more parameters in Table 1) when the step size is increased by a factor of 5, leading to less data. The mean error is $0.20\%$ and the median error is $0.16\%$, using an artificial neural network with 7 hidden layers with 40 neurons per layer.

Figure 14

Error histograms of ${\alpha }_1$ (left) and ${\alpha }_2$ (right) for two scenarios. Top panels: Using radiographs generated by proton beams in the $z$ direction as training data, the mean errors are $1.20\times {10}^{-2}\%$ and $3.48\%$ while the median errors are $3.59\times {10}^{-3}\%$ and $2.50\%$ for ${\alpha }_1$ and ${\alpha }_2$, respectively, using an artificial neural network with 1 hidden layer consisting of 30 neurons (more parameters in Table 1). The errors for ${\alpha }_2$ (field strength of the flux rope) are high because the field structure associated with ${\alpha }_1$ (ellipsoidal blob) is deflecting many protons away from the flux rope, causing a lack of information in the resulting radiographs. Bottom panels: Using radiographs generated by proton beams in the $z$ and $x$ directions as training data, the mean errors are $1.81\times {10}^{-4}\%$ and $0.11\%$ while the median errors are $1.41\times {10}^{-4}\%$ and $8.68\times {10}^{-2}\%$ for ${\alpha }_1$ and ${\alpha }_2$, respectively, using an artificial neural network with 1 hidden layer consisting of 110 neurons. We see that upon including data from the proton beam in the $x$ direction, more information for both field structures is added to the data set and errors reduce by at least an order of magnitude.