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.

Learning meaningful representations of complex objects that can be seen through multiple ($k\geq 3$) views or modalities is a core task in machine learning. Existing methods use losses originally intended for paired views, and extend them to $k$ views, either by instantiating $\tfrac12k(k-1)$ loss-pairs, or by using reduced embeddings, following a *one vs. average-of-rest* strategy. We propose the multi-marginal matching gap (M3G), a loss that borrows tools from multi-marginal optimal transport (MM-OT) theory to simultaneously incorporate all $k$ views. Given a batch of $n$ points, each seen as a $k$-tuple of views subsequently transformed into $k$ embeddings, our loss contrasts the cost of matching these $n$ ground-truth $k$-tuples with the MM-OT polymatching cost, which seeks $n$ optimally arranged $k$-tuples chosen within these $n\times k$ vectors. While the exponential complexity $O(n^k$) of the MM-OT problem may seem daunting, we show in experiments that a suitable generalization of the Sinkhorn algorithm for that problem can scale to, e.g., $k=3\sim 6$ views using mini-batches of size $64~\sim128$. Our experiments demonstrate improved performance over multiview extensions of pairwise losses, for both self-supervised and multimodal tasks.