The ability to learn polynomials and generalize out-of-distribution is essential for simulation metamodels in many disciplines of engineering, where the time step updates are described by polynomials. While feed forward neural networks can fit any function, they cannot generalize out-of-distribution for higher-order polynomials. Therefore, this paper collects and proposes multiplicative neural network (MNN) architectures that are used as recursive building blocks for approximating higher-order polynomials. Our experiments show that MNNs are better than baseline models at generalizing, and their performance in validation is true to their performance in out-of-distribution tests. In addition to MNN architectures, a simulation metamodeling approach is proposed for simulations with polynomial time step updates. For these simulations, simulating a time interval can be performed in fewer steps by increasing the step size, which entails approximating higher-order polynomials. While our approach is compatible with any simulation with polynomial time step updates, a demonstration is shown for an epidemiology simulation model, which also shows the inductive bias in MNNs for learning and generalizing higher-order polynomials.

翻译：在工程学众多领域中，仿真元模型需具备学习多项式并泛化分布外数据的能力，因为其时步更新由多项式描述。前馈神经网络虽能拟合任意函数，但无法对高阶多项式实现分布外泛化。为此，本文收集并提出了乘法神经网络架构，该架构可作为递归构建模块用于逼近高阶多项式。实验表明，MNN在泛化性能上优于基线模型，且在验证集与分布外测试中的表现高度一致。除MNN架构外，本文还针对时步更新为多项式形式的仿真提出了一种仿真元建模方法。在此类仿真中，通过增大步长可减少模拟时间区间所需的步数，而这需要逼近高阶多项式。尽管我们的方法适用于任何含多项式时步更新的仿真，但本文以流行病学仿真模型为例进行了验证，该案例同时展示了MNN在学习与泛化高阶多项式中的归纳偏置。