TRENDy：有效非线性动力学的时间回归
TRENDy: Temporal Regression of Effective Nonlinear Dynamics
AI总结：
TRENDy提出一种无方程方法，通过多尺度滤波和神经ODE学习时空动力学的低维有效模型，能自动定位Turing和Hopf分岔，并成功应用于蜥蜴发育图案分析，揭示表面几何对反应-扩散机制的影响。
摘要：
时空动力学普遍存在于自然科学中，从控制动物色素沉着图案的形态发生动力学到调控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，这一任务尤为困难，因为控制偏微分方程（PDEs）未知，且数据有限且含有噪声。为应对这一挑战，我们提出了TRENDy（有效非线性动力学的时间回归），一种无方程方法，用于学习时空动力学的低维预测模型。TRENDy首先通过多尺度滤波操作的级联将输入数据映射到有效动力学的低维空间。我们的关键洞见在于认识到这些有效动力学可以通过一个具有与输入PDE相同参数空间的神经常微分方程（NODE）来拟合。前述滤波操作强烈正则化了NODE的相空间，使得TRENDy相比现有方法对噪声具有显著更强的鲁棒性。我们训练TRENDy来预测代表物理和生命科学中各种动力学的合成数据及真实数据的有效动力学。随后，我们展示了如何自动定位参数空间中未见区域的Turing和Hopf分岔。最后，我们将该方法应用于蜥蜴（ocellated lizard）发育过程中空间图案形成的分析。我们发现，TRENDy预测的有效状态不仅准确预测了随时间的空间变化，还识别出不同解剖区域（如尾部、颈部、身体）特有的不同图案特征——这一洞见凸显了表面几何对反应-扩散机制及其在驱动空间变化图案动力学中作用的潜在影响。
英文摘要：
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 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. TRENDy first maps input data to a low-dimensional space of effective dynamics through 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 we 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 predicted effective state not only accurately predicts spatial changes over time but also identifies distinct pattern features unique to different anatomical regions, such as the tail, neck, and body--an insight that highlights the potential influence of surface geometry on reaction-diffusion mechanisms and their role in driving spatially varying pattern dynamics.
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