Projection-based reduced order models are effective at approximating parameter-dependent differential equations that are parametrically separable. When parametric separability is not satisfied, which occurs in both linear and nonlinear problems, projection-based methods fail to adequately reduce the computational complexity. Devising alternative reduced order models is crucial for obtaining efficient and accurate approximations to expensive high-fidelity models. In this work, we develop a time-stepping procedure for dynamical parameter-dependent problems, in which a neural-network is trained to propagate the coefficients of a reduced basis expansion. This results in an online stage with a computational cost independent of the size of the underlying problem. We demonstrate our method on several parabolic partial differential equations, including a problem that is not parametrically separable.

翻译：以预测为基础的降序模型对于接近以参数为依存的差异方程式是有效的,这些方程式是可分的。当参数分离性无法满足时(这发生在线性和非线性问题中),以预测为基础的方法无法充分减少计算的复杂性。设计替代的降序模型对于获得价格昂贵的高不忠模型的高效和准确近似值至关重要。在这项工作中,我们为动态的、以参数为依存的问题制定了一个时间跨步程序,其中对神经网络进行了培训,以传播减少基数扩展的系数。这导致在网上阶段出现计算成本,而计算成本与根本问题的规模无关。我们用几种参数偏差部分方程式演示了我们的方法,包括一个非分立性分离的问题。