Proton radiography is an effective technique for diagnosing field distributions in plasmas. However, due to the complexity of electromagnetic field structures, reconstructing electromagnetic fields from proton radiographs is extremely challenging and often requires some simplified symmetry assumptions about the fields. Here, we present a machine learning approach to reconstruct three-dimensional (3D) magnetic field distributions from complex proton radiographs without relying on such assumptions.
To enable this, we construct the target 3D magnetic fields by linearly superposing multiple elementary magnetic structures generated from the Weibel instability. Each element is characterized by eight parameters—structural parameters (a, b, B₀), spatial coordinates (x₀, y₀, z₀), and rotation angles (θ, ϕ)—resulting in 80 degrees of freedom in total. Parameters were uniformly sampled within ±25% of their baseline values, and a dataset of 50,000 magnetic field–proton radiograph pairs was generated through forward simulation using GEANT4. All proton radiographs reside in the caustic regime, exhibiting multiple asymmetric caustics and significant flux concentrations.
A lightweight three-layer convolutional neural network (CNN) was designed for the reconstruction task. The network consists of an input layer, three convolutional modules (the first two following a ”convolution–batch normalization–max pooling” cascaded structure, and the third is simplified to a single convolutional layer), a flattening layer, a dropout layer, and an output layer. Bayesian optimization was applied to determine the optimal hyperparameters. The model was trained on 40000 samples, with 5000 samples for validation and 5000 for testing.
On the test set, the CNN achieves a mean absolute percentage error (MAPE) of 8.5% in predicting the 80 magnetic parameters, below the 12.9% random-guessing threshold. Prediction errors for most parameters follow near-zero-mean Gaussian distributions, with relative standard deviations under 6%. The reconstructed fields show high spatial agreement with the reference fields, and corresponding proton images match the originals with a cosine similarity of 0.89.
This study demonstrates that our CNN-based proton radiography reconstruction method can effectively reconstruct complex 3D magnetic fields without symmetry assumptions or manual parameter tuning, offering a novel tool for diagnosing electromagnetic fields in high-intensity laser-plasma interactions. Future work may incorporate multi-angle proton radiography and transfer learning from experimental data to enhance the method’s practicality and robustness.