We present a new algorithm for learning unknown governing equations from trajectory data, using and ensemble of neural networks. Given samples of solutions $x(t)$ to an unknown dynamical system $\dot{x}(t)=f(t,x(t))$, we approximate the function $f$ using an ensemble of neural networks. We express the equation in integral form and use Euler method to predict the solution at every successive time step using at each iteration a different neural network as a prior for $f$. This procedure yields M-1 time-independent networks, where M is the number of time steps at which $x(t)$ is observed. Finally, we obtain a single function $f(t,x(t))$ by neural network interpolation. Unlike our earlier work, where we numerically computed the derivatives of data, and used them as target in a Lipschitz regularized neural network to approximate $f$, our new method avoids numerical differentiations, which are unstable in presence of noise. We test the new algorithm on multiple examples both with and without noise in the data. We empirically show that generalization and recovery of the governing equation improve by adding a Lipschitz regularization term in our loss function and that this method improves our previous one especially in presence of noise, when numerical differentiation provides low quality target data. Finally, we compare our results with the method proposed by Raissi, et al. arXiv:1801.01236 (2018) and with SINDy.
翻译:我们提出了一个新的算法,从轨迹数据中学习未知的正方程式,使用和混合神经网络,从轨迹数据中学习不为人知的正方程式。根据解决方案的样本 $x(t) 美元到未知的动态系统 $\d{x}(t)=f(t,x(t))美元,我们使用神经网络的混合体来估计函数 $f(t,x(t)36) 美元。我们用整体形式表达方程式,并使用 Euler 方法在每一个连续步骤中用不同神经网络的循环来预测解决方案。这个程序产生 M-1 时间依赖网络, M 是观察到美元(t) 美元(t) 美元) 的不为未知的动态系统 。 最后,我们通过神经网络的内化网络的内推法,我们用数字来计算数据衍生的衍生物, 并且用Lipschitz固定的神经网络作为目标, 我们的新方法避免了数字差异, 在噪音出现时,我们用一个新的算法来测试一个新的算法, 既要用以前的正值来改善我们的标准, 也用直方程式来显示我们的数据。