We present a combined numerical and data-driven workflow for efficient prediction of nonlinear, instationary convection-diffusion-reaction dynamics on a two-dimensional phenotypic domain, motivated by macroscopic modeling of cancer cell plasticity. A finite-difference solver, implemented in C++, is developed using second-order spatial discretizations and a step-size controlled Runge-Kutta time integrator. A mesh refinement study confirms the second-order convergence for the spatial discretizations error. Based on simulated input-output pairs and corresponding parameterizations for the diffusion, advection, and reaction mechanisms, we train a parameter-conditioned U-Net surrogate to approximate the fixed-horizon solution map. The surrogate incorporates Feature-wise Linear Modulation (FiLM) for parameter conditioning, coordinate encoding to incorporate spatial location information, and residual blocks to enable multiscale representation learning in combination with the U-Nets skip connections. The trained model achieves low prediction error on held-out test data and provides favorable prediction times due to the GPU based parallelization. Generalization is analyzed using a factorial test dataset, separating initial conditions from parameter conditioning. The results reveal that approximation difficulty varies primarily with the conditioning vector (i.e., the induced PDE regime), rather than with the initial conditions.

翻译：本文提出一种数值与数据驱动相结合的工作流程，用于高效预测二维表型域上的非线性非定常对流-扩散-反应动力学，其研究动机源于癌细胞可塑性的宏观建模。我们基于二阶空间离散格式和步长控制龙格-库塔时间积分器，开发了C++实现的有限差分求解器。网格细化研究证实了空间离散误差的二阶收敛性。基于模拟的输入-输出对及其对应的扩散、平流和反应机制参数化设置，我们训练了一个参数化条件U-Net代理模型以逼近固定时间范围的解映射。该代理模型融合了三种关键技术：采用特征线性调制（FiLM）实现参数条件编码，通过坐标编码融入空间位置信息，并利用残差块结合U-Net跳跃连接实现多尺度表征学习。训练后的模型在保留测试集上实现了较低的预测误差，且基于GPU并行化获得了优越的预测速度。通过析因测试数据集（分离初始条件与参数条件）进行泛化能力分析，结果表明近似难度主要随条件向量（即诱导的偏微分方程机制）变化，而非初始条件。