PolyNODE: Variable-dimension Neural ODEs on M-polyfolds

We introduce steerable neural ordinary differential equations on homogeneous spaces $M=G/H$. These models constitute a novel geometric extension of manifold neural ordinary differential equations (NODEs) that transport associated feature vectors transforming under the local symmetry group $H$. We interpret features as sections of associated vector bundles over $M$, and describe their evolution as parallel transport. This results in a coupled system of ODEs consisting of a flow equation on $M$ and a steering equation acting on features. We show that steerable NODEs are $G$-equivariant whenever the vector field generating the flow and the connection governing parallel transport are both $G$-invariant. Furthermore, we demonstrate how steerable NODEs incorporate existing NODE models and continuous normalizing flows on Lie groups. Our framework provides the geometric foundation for learning continuous-time equivariant dynamics of general vector-valued features on homogeneous spaces.

We introduce Tadpole, a novel foundation model for three-dimensional partial differential equations (PDEs) that addresses key challenges in transferability, scalability to high dimensionality, and multi-functionality. Tadpole is pre-trained as an autoencoder on synthetic 3D PDE data generated by an efficient online data-generation framework. This enables large-scale, diverse training without storage or I/O overhead, demonstrated by scaling to an equivalent of hundreds of terabytes of training data. By autoencoding single-channel spatial crops, Tadpole learns rich and transferable representations across heterogeneous physical systems with varying numbers of state variables and spatial resolutions. Although pre-trained solely as an autoencoder, Tadpole can be efficiently applied for multiple downstream tasks beyond reconstruction, including dynamics learning and generative modeling. For dynamics learning, we propose a novel parameter-efficient fine-tuning strategy that integrates low-rank adaptation, latent-space transformations, and reintroduced skip connections, achieving accurate temporal modeling with a minimal number of trainable parameters. Tadpole demonstrates strong fine-tuning performance across various downstream tasks, highlighting its versatility and effectiveness as a foundation model for 3D PDE learning. Source code and pre-trained weights of Tadpole are available at https://github.com/tum-pbs/tadpole

We show that using autonomous-flow-based generation, one can universally approximate orientation-preserving diffeomorphisms defined on the cube by Neural ODEs with rate $\mathcal{O}(P^{-1/d})$ with $P$ parameters. On the other hand, we show that by using only a single autonomous flow, the class of Neural ODEs is nowhere dense on the cube in dimension $d \ge 2$ . Under a compact-support$_\mathrm{id}$ condition on $(0,1)^d$, we show that using autonomous-flow-based generation, one can universally approximate compactly supported$_\mathrm{id}$ diffeomorphisms on $(0,1)^d$ for any dimension with rate $\mathcal{O}((\frac{P}{\log P})^{-2/d})$ with $P$ parameters and for compactly supported$_\mathrm{id}$ homeomorphisms on $(0,1)^d$ in dimension $d \geq 5$ with rate $\mathcal{O}(P^{-1/(d+1)})$ with $P$ parameters and by a composition of at most $I_d$ autonomous Neural ODEs with the same support$_\mathrm{id}$, where $I_d$ depends only on the dimension. Moreover, we show that the class of single autonomous flows compactly supported$_\mathrm{id}$ on $(0,1)^d$ is meagre in the space of compactly supported$_\mathrm{id}$ homeomorphisms on $(0,1)^d$ for $d\ge 2$. By linearly lifting the domain into one higher dimension, we obtain a universal approximation result for Lipschitz functions compactly supported on $(0,1)^d$ with rate $\mathcal{O}(P^{-1/(d+1)})$ with $P$ parameters.