In many mechanistic medical, biological, physical and engineered spatiotemporal dynamic models the numerical solution of partial differential equations (PDEs) can make simulations impractically slow. Biological models require the simultaneous calculation of the spatial variation of concentration of dozens of diffusing chemical species. Machine learning surrogates, neural networks trained to provide approximate solutions to such complicated numerical problems, can often provide speed-ups of several orders of magnitude compared to direct calculation. PDE surrogates enable use of larger models than are possible with direct calculation and can make including such simulations in real-time or near-real time workflows practical. Creating a surrogate requires running the direct calculation tens of thousands of times to generate training data and then training the neural network, both of which are computationally expensive. We use a Convolutional Neural Network to approximate the stationary solution to the diffusion equation in the case of two equal-diameter, circular, constant-value sources located at random positions in a two-dimensional square domain with absorbing boundary conditions. To improve convergence during training, we apply a training approach that uses roll-back to reject stochastic changes to the network that increase the loss function. The trained neural network approximation is about 1e3 times faster than the direct calculation for individual replicas. Because different applications will have different criteria for acceptable approximation accuracy, we discuss a variety of loss functions and accuracy estimators that can help select the best network for a particular application.
翻译:在许多机械式医学、生物、物理和工程的时空动态模型中,部分差异方程式(PDEs)的数值解决方案可以使模拟不切实际地缓慢地进行模拟。生物模型要求同时计算数十种挥发化学物种集中的空间变异。机器学习代孕器、经过训练为如此复杂的数字问题提供近似解决办法的神经网络,往往能够提供与直接计算相比数个数量级的加速增量。PDE代孕器能够使用比直接计算可能使用的更大数量级的模型,并且可以使在实时或近实时工作流程中包括这种模拟变得实用。建立代理模型需要直接计算数万次,以生成培训数据,然后对神经网络进行培训,两者在计算成本上都是昂贵的。我们使用演进神经网络来将固定式解决方案与在两个等距、循环、定值来源的随机位置,吸收边界条件。为了改进培训过程中的趋同,我们采用了一种培训方法,即直接计算准确性方法,要用数以数以千计数计数计数,然后对一个经过训练的网络进行反向的精确度调整的计算。