Modeling continuous dynamical systems from discretely sampled observations is a fundamental problem in data science. Often, such dynamics are the result of non-local processes that present an integral over time. As such, these systems are modeled with Integro-Differential Equations (IDEs); generalizations of differential equations that comprise both an integral and a differential component. For example, brain dynamics are not accurately modeled by differential equations since their behavior is non-Markovian, i.e. dynamics are in part dictated by history. Here, we introduce the Neural IDE (NIDE), a novel deep learning framework based on the theory of IDEs where integral operators are learned using neural networks. We test NIDE on several toy and brain activity datasets and demonstrate that NIDE outperforms other models. These tasks include time extrapolation as well as predicting dynamics from unseen initial conditions, which we test on whole-cortex activity recordings in freely behaving mice. Further, we show that NIDE can decompose dynamics into their Markovian and non-Markovian constituents via the learned integral operator, which we test on fMRI brain activity recordings of people on ketamine. Finally, the integrand of the integral operator provides a latent space that gives insight into the underlying dynamics, which we demonstrate on wide-field brain imaging recordings. Altogether, NIDE is a novel approach that enables modeling of complex non-local dynamics with neural networks.
翻译:从分散抽样的观测中模拟连续动态系统是数据科学的一个根本问题。通常,这种动态是非本地过程的结果,随着时间的推移形成一个整体。因此,这些系统与Integro-Differential Equations(IDES)建模;包含一个整体和差异组成部分的差别方程式的概观。例如,大脑动态并不是用差异方程式准确建模,因为其行为是非马尔科米亚的,即动态部分由历史决定。在这里,我们引入了Neoral IDE(NIDE),这是一个基于IDE理论的新颖的非深层次学习框架,其基础是综合操作者利用神经网络学习综合操作者。我们用若干微小和脑活动数据集测试NIDE,并表明NIDE优于其他模型。这些任务包括时间外推以及从隐蔽初始条件下预测动态,我们测试的是全层层结构模型在自由演化的老鼠中记录。我们展示了NIDE将动态引入其新的内脏和非马尔科维(NI)的内深非内隐性网络,通过智能网络,我们通过学习的综合操作者将一个基础的智能操作者展示了大脑图像。