Spatiotemporal dynamics pervade the natural sciences, from the morphogen dynamics underlying patterning in animal pigmentation to the protein waves controlling cell division. A central challenge lies in understanding how controllable parameters induce qualitative changes in system behavior called bifurcations. This endeavor is made particularly difficult in realistic settings where governing partial differential equations (PDEs) are unknown and data is limited and noisy. To address this challenge, we propose TRENDy (Temporal Regression of Effective Nonlinear Dynamics), an equation-free approach to learning low-dimensional, predictive models of spatiotemporal dynamics. Following classical work in spatial coarse-graining, TRENDy first maps input data to a low-dimensional space of effective dynamics via a cascade of multiscale filtering operations. Our key insight is the recognition that these effective dynamics can be fit by a neural ordinary differential equation (NODE) having the same parameter space as the input PDE. The preceding filtering operations strongly regularize the phase space of the NODE, making TRENDy significantly more robust to noise compared to existing methods. We train TRENDy to predict the effective dynamics of synthetic and real data representing dynamics from across the physical and life sciences. We then demonstrate how our framework can automatically locate both Turing and Hopf bifurcations in unseen regions of parameter space. We finally apply our method to the analysis of spatial patterning of the ocellated lizard through development. We found that TRENDy's effective state not only accurately predicts spatial changes over time but also identifies distinct pattern features unique to different anatomical regions, highlighting the potential influence of surface geometry on reaction-diffusion mechanisms and their role in driving spatially varying pattern dynamics.

翻译：时空动力学普遍存在于自然科学中，从动物色素沉着模式背后的形态发生素动力学，到控制细胞分裂的蛋白质波。一个核心挑战在于理解可控参数如何引发系统行为的定性变化（即分岔）。在现实场景中，由于控制偏微分方程未知且数据有限且含有噪声，这一研究变得尤为困难。为应对此挑战，我们提出了TRENDy（有效非线性动力学的时间回归），这是一种无方程方法，用于学习时空动力学的低维预测模型。借鉴空间粗粒化的经典工作，TRENDy首先通过一系列多尺度滤波操作将输入数据映射到有效动力学的低维空间。我们的关键见解是认识到这些有效动力学可以通过一个与输入偏微分方程具有相同参数空间的神经常微分方程来拟合。前述滤波操作对神经常微分方程的相空间进行了强正则化，使得TRENDy与现有方法相比对噪声具有显著更强的鲁棒性。我们训练TRENDy来预测合成及真实数据的有效动力学，这些数据代表了物理和生命科学领域的各种动力学过程。随后，我们展示了我们的框架如何能自动定位参数空间未见区域中的图灵分岔和霍普夫分岔。最后，我们将该方法应用于分析眼斑蜥蜴发育过程中的空间模式形成。我们发现，TRENDy的有效状态不仅能准确预测空间随时间的变化，还能识别不同解剖区域特有的独特模式特征，这突显了表面几何形状对反应-扩散机制的潜在影响及其在驱动空间变化模式动力学中的作用。