Objective: Real-time adaptive proton range verification systems based on produced neutrons require accurate information on their non-isotropic momentum distributions within short times, for which Monte Carlo (MC) methods are too computationally expensive. We present a surrogate model based on Fourier Neural Operators (FNO) for fast prediction of angle- and energy-resolved proton transport and neutron production within proton therapy. Approach: We treat the irradiated phantom and the proton beam's state as depth-evolving series, respectively of different materials, and of spatial, angular and energy phase space density distributions. The task is solved auto-regressively by learning changes in the distributions of protons and those of produced neutrons. For training and evaluation, two datasets of 47 MC simulations featuring different primary intensities were produced. Simulated geometries were extracted from a thoracic CT scan as series of laterally homogeneous materials. Main Results: An average relative $L^2$ discrepancy of 0.067 and 0.137 was achieved by the predicted proton and neutron distributions, respectively. This corresponded to an average gamma passing rate in the spatial distributions of 99.95$\%$ and 99.40$\%$. Training with higher primary intensities led to improvements between 12$\%$ and 30$\%$ in density metrics. Inference over depths of 40 cm at a resolution of 0.5 mm required on average 23.17 s per beam. Significance: The proposed proton beam surrogate generates accurate spatial and momentum distributions of neutrons at MC-level accuracy within seconds, while demonstrating robust generalization with respect to irradiated geometry and beam characteristics. This approach can be used for prototyping and operation of range verification systems, other tasks such as neutron dose estimation, and can be extended to include other kinds of secondary emissions.

翻译：目的：基于产生中子的实时自适应质子射程验证系统，需要在短时间内获得其中子非各向同性动量分布的准确信息，而蒙特卡洛（MC）方法计算成本过高。我们提出了一种基于傅里叶神经算子（FNO）的代理模型，用于快速预测质子治疗中角度与能量分辨的质子输运及中子产生。方法：我们将辐照体模和质子束状态分别视为深度演化的序列，前者由不同材料构成，后者为空间、角度和能量相空间密度分布。该任务通过自回归学习质子分布及产生中子分布的变化来解决。为进行训练和评估，我们生成了两个包含47次MC模拟的数据集，这些模拟具有不同的初级粒子强度。模拟几何结构从胸部CT扫描中提取，作为一系列横向均匀的材料序列。主要结果：预测的质子分布和中子分布分别实现了0.067和0.137的平均相对$L^2$误差。这对应于空间分布中99.95$\%$和99.40$\%$的平均伽马通过率。使用更高初级粒子强度进行训练，使得密度指标提升了12$\%$至30$\%$。在0.5 mm分辨率下对40 cm深度进行推断，平均每束流需时23.17秒。意义：所提出的质子束代理模型能在数秒内以MC级别的精度生成准确的中子空间与动量分布，同时在辐照几何和束流特性方面展现出稳健的泛化能力。该方法可用于射程验证系统的原型设计与运行、中子剂量估计等其他任务，并可扩展至包含其他类型的次级发射。