cs.LGFeb 16, 2026

PolyNODE: Variable-dimension Neural ODEs on M-polyfolds

Authors: Per Åhag, Alexander Friedrich, Fredrik Ohlsson, Viktor Vigren Näslund

Organizations: Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden

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Abstract

Neural ordinary differential equations (NODEs) are geometric deep learning models based on dynamical systems and flows generated by vector fields on manifolds. Despite numerous successful applications, particularly within the flow matching paradigm, all existing NODE models are fundamentally constrained to fixed-dimensional dynamics by the intrinsic nature of the manifold's dimension. In this paper, we extend NODEs to M-polyfolds (spaces that can simultaneously accommodate varying dimensions and a notion of differentiability) and introduce PolyNODEs, the first variable-dimensional flow-based model in geometric deep learning. As an example application, we construct explicit M-polyfolds featuring dimensional bottlenecks and PolyNODE autoencoders based on parametrised vector fields that traverse these bottlenecks. We demonstrate experimentally that our PolyNODE models can be trained to solve reconstruction tasks in these spaces, and that latent representations of the input can be extracted and used to solve downstream classification tasks. The code used in our experiments is publicly available at https://github.com/turbotage/PolyNODE .

Figures & tables

Figure 1: Illustration (grey shading) of the stratified topological space Ω11⊂R3\Omega^{1}_{1}\subset\R^{3} with τ1=−2\tau_{1}=-2 and τ2=2\tau_{2}=2 .
Figure 2: Illustration of a semi-flow in Ω11⊂R3\Omega^{1}_{1}\subset\R^{3} with τ1=−2\tau_{1}=-2 and τ2=2\tau_{2}=2 , polyfold structure indicated as grey planes.
Figure 3: Illustration of a flow line entering the bottleneck at τ1\tau_{1} and exiting at τ2\tau_{2} . Explicit compression starts at τ0\tau_{0} .
Figure 4: (a) Flow lines (green) for individual input samples for N=1N=1 . Input set X\mathcal{X} (blue) and reconstructed output Y^\hat{\mathcal{Y}} (orange). The y2y_{2} component is projected out for visualisation. (b) Time slices of the flow for a spiral with N=4N=4 . Colour scale corresponds to the angular parameter.
Figure 5: Relative monotonicity error during training for N=4N=4 .
Figure 6: Time slices of the encoding (a) and decoding (b) of the 22 -sphere in R4\R^{4} . The colour represents the coordinate in the fourth dimension.
Figure 7: Classifier flow lines of the radial (a) and angular (b) classification problems. Colours indicate which label the trajectory belongs to and what label the target points (squares) correspond to. The latent representations constituting the classifier input correspond to the line where the integral curves start.
Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

RegionShape of Vector FieldNeural NetworksInput
{τ<τ0}\{\tau<\tau_{0}\}(C,Y2,...,Yn+m+1)(C,Y_{2},...,Y_{n+m+1}){Y}2n+m+1\{Y\}_{2}^{n+m+1}(τ,x,y)(\tau,x,y)
{τ0<τ<τ1}\{\tau_{0}<\tau<\tau_{1}\}(C,0,...,0,Yn+2,...Yn+m+1)(C,0,...,0,Y_{n+2},...Y_{n+m+1}) , {Yi}i=n+2n+m+1\{Y_{i}\}_{i=n+2}^{n+m+1} as in ( 8 ),-(τ,x,y)(\tau,x,y)
{τ1<τ<τ2}\{\tau_{1}<\tau<\tau_{2}\}(C,0,...,0)(C,0,...,0)--
{τ>τ2}\{\tau>\tau_{2}\}(C,Y2,...,Yn+m+1)(C,Y_{2},...,Y_{n+m+1}){Y}2n+m+1\{Y\}_{2}^{n+m+1}(τ,x,y)(\tau,x,y)
Appendix
Table 1: Structure of vector fields for reconstruction experiments
Figure 8: Unwinding and encoding of spirals with number of turns ranging from N=2N=2 to N=5N=5
Figure 9: Reconstruction (2d projection) of spirals with number of turns ranging from N=2N=2 to N=5N=5
Figure 10: Mean euclidean reconstruction error relative to extrinsic spiral diameter
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