TRENDy: Temporal Regression of Effective Nonlinear Dynamics

TL;DR

TRENDy addresses the challenge of learning predictive, parametric effective dynamics from noisy spatiotemporal PDE data without requiring full equation recovery. It maps high-dimensional PDE states to a low-dimensional effective space using the scattering transform and fits a neural ODE whose parameters align with the PDE’s parameter space, enabling bifurcation-aware predictions. The approach is validated on Gray Scott and Brusselator models and demonstrated on ocellated lizard patterning, showing automatic detection of Turing and Hopf bifurcations and interpretable links between geometry and pattern dynamics. TRENDy offers a robust, equation-free framework for analyzing complex spatiotemporal systems in biology and physics with real-world data.

Abstract

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.

Figures (12)

• Figure 1: TRENDy. (Left) Observed dynamics are solutions to PDEs, $u_t = N[u(x,y); \theta]$, with parameters $\theta$ and states $u(x,y)$ taking values on a square domain $\Omega\subset \mathbb{R}^2$. Spatial features are measured at each $t$ using recursive, multiscale filtering via scattering, $\Phi$ (see Secs. \ref{['sec:methods']}, \ref{['app:scattering']}), and stacked into a single representation, ($\mathcal{S}^{0}(t) : \mathcal{S}^{1}(t) : \mathcal{S}^{2}(t), \ldots$), where superscripts represent the order of recursion. $\Phi$ thus maps $u$ to reduced-order trajectories which can be further reduced by ranking and subsampling (red). This "effective" representation is controlled by unknown temporal dynamics, $\dot{a}$. (Middle) Those unknown dynamics are modeled as a neural network, $g_{\pi}$, having learnable weights, $\pi$, and which depend on the known, true parameters, $\theta$. (Right) Simulated trajectories from the estimated effective dynamics, $\tilde{a}(t)$, are initialized on the true $a(0)$ (using the initial scattering coefficients $\mathcal{S}^{i}(0)$) and regressed against true effective trajectories, $a(t)$, with a pointwise loss, $L$.
• Figure 2: Bifurcation prediction in the GS model with TRENDy. (Left): the bifurcation landscape. The GS model transitions to patterning near the orange curve in $F-k$ space. Under the curve, the system is spatially homogeneous ($u\approx 1$, yellow inset). Near the curve, it produces a wide variety of patterns (stripe inset). After the curve, the system is homogeneous again ($u \approx 0$, teal inset). Training data was taken from a thin region of width $.01$ and centered on $F=.054$. A rectangle of height $.01$ was held out as test data centered on the bifurcation value of $k^*=.062$. (Right): Detection performance. Numerical continuation was performed on TRENDy trained on five different measurements plus SINDyCP (SG = spatial gradient, T$d$ = TRENDy with $d$ coefficients, FV=Fourier vector, SCP=SINDyCP; details in main text). Estimated $k^*$ values (y axis) are plotted for each of the three noise conditions (Clean as circles, Boundaries as squares and Patches as triangles) for all models. TRENDy with 10 scattering coefficients had the best performance and noise degradation. See main text for details.
• Figure 3: Predicting effective patterning dynamics. (Left): true patterning landscape. Samples were generated from the test region shown in Fig. \ref{['fig:gs_bif']} and classified with 4-way k-means using all scattering coefficients (8 orientations, 6 scales, up to order 2). K-means classes (example in insets) were (1) dense spots, (2) homogeneity, (3) sparse spots, and (4) stripes. (Right): A 4-way support vector machine was trained on the predicted state of TRENDy using five types of measurements (see main text). A theoretical maximum F1 score was computed by classifying all features used to generate the original labels (dashed line). F1 scores on test data in each condition are plotted.
• Figure 4: TRENDy learns reduced-order models of the Brusselator.(Top row) TRENDy was trained on zeroth-order scattering coefficients, ($S_{1}$ in solid green, $S_{2}$ in solid orange) measured from the evolving Brusselator model (Eq. \ref{['eq:brusselator']}). The trained model (here depicted with hold-out parameter $\epsilon=.15$ and without noise) closely predicted the true effective dynamics across the bifurcation boundary (depicted: $A=1.5$, $B=1.5, 3.0, 4.0$). (Bottom row). For each $A$ between 1.5 and 3.5 in 100 steps, numerical continuation was performed on the trained TRENDy model. Whenever a Hopf bifurcation was detected at $B$, that point $(A,B)$ was tabulated and plotted. Then, a quadratic polynomial was fit to all points within one holdout ($\epsilon$) and noise condition. The true Hopf manifold is in orange at $B=1+A^2$. Approximation of the manifold is qualitatively correct across conditions, but worsens in the boundaries and patches conditions.
• Figure 5: TRENDy learns scale dynamics of the ocellated lizard.(Left, top inset). A cropped version of the adult ocellated lizard's torso in medio-lateral, cranio-caudal coordinates. Four regions are highlighted in colors corresponding to their quadrant label in this coordinate system. (Left, lower). Dynamics of one patch of scales taken from the origin from juvenile to adult state. Features represented spatially averaged scattering coefficients. Solid lines are the true dynamics and dashed are TRENDy estimates. The insets depict the patch at $t=0, 125, 250$. (Right, top inset). An SVM was trained on the final, 10-d state of TRENDy with labels given by the patch's quadrant. This inset depicts prediction on the test data, where TRENDy achieved 95$\%$ accuracy. (Right, lower). Together with the classifier results, the moderate clustering of TRENDy states in PCA space indicates a relation between scale dynamics and anatomical coordinates.