Many patterns in nature exhibit self-similarity: they can be compactly described via self-referential transformations. Said patterns commonly appear in natural and artificial objects, such as molecules, shorelines, galaxies and even images. In this work, we investigate the role of learning in the automated discovery of self-similarity and in its utilization for downstream tasks. To this end, we design a novel class of implicit operators, Neural Collages, which (1) represent data as the parameters of a self-referential, structured transformation, and (2) employ hypernetworks to amortize the cost of finding these parameters to a single forward pass. We investigate how to leverage the representations produced by Neural Collages in various tasks, including data compression and generation. Neural Collages image compressors are orders of magnitude faster than other self-similarity-based algorithms during encoding and offer compression rates competitive with implicit methods. Finally, we showcase applications of Neural Collages for fractal art and as deep generative models.

We detail a novel class of implicit neural models. Leveraging time-parallel methods for differential equations, Multiple Shooting Layers (MSLs) seek solutions of initial value problems via parallelizable root-finding algorithms. MSLs broadly serve as drop-in replacements for neural ordinary differential equations (Neural ODEs) with improved efficiency in number of function evaluations (NFEs) and wall-clock inference time. We develop the algorithmic framework of MSLs, analyzing the different choices of solution methods from a theoretical and computational perspective. MSLs are showcased in long horizon optimal control of ODEs and PDEs and as latent models for sequence generation. Finally, we investigate the speedups obtained through application of MSL inference in neural controlled differential equations (Neural CDEs) for time series classification of medical data.

Effective control and prediction of dynamical systems require appropriate handling of continuous-time and discrete, event-triggered processes. Stochastic hybrid systems (SHSs), common across engineering domains, provide a formalism for dynamical systems subject to discrete, possibly stochastic, state jumps and multi-modal continuous-time flows. Despite the versatility and importance of SHSs across applications, a general procedure for the explicit learning of both discrete events and multi-mode continuous dynamics remains an open problem. This work introduces Neural Hybrid Automata (NHAs), a recipe for learning SHS dynamics without a priori knowledge on the number, mode parameters, and inter-modal transition dynamics. NHAs provide a systematic inference method based on normalizing flows, neural differential equations, and self-supervision. We showcase NHAs on several tasks, including mode recovery and flow learning in systems with stochastic transitions, and end-to-end learning of hierarchical robot controllers.

Continuous deep learning architectures have recently re-emerged as Neural Ordinary Differential Equations (Neural ODEs). This infinite-depth approach theoretically bridges the gap between deep learning and dynamical systems, offering a novel perspective. However, deciphering the inner working of these models is still an open challenge, as most applications apply them as generic black-box modules. In this work we "open the box", further developing the continuous-depth formulation with the aim of clarifying the influence of several design choices on the underlying dynamics.

The infinite-depth paradigm pioneered by Neural ODEs has launched a renaissance in the search for novel dynamical system-inspired deep learning primitives; however, their utilization in problems of non-trivial size has often proved impossible due to poor computational scalability. This work paves the way for scalable Neural ODEs with time-to-prediction comparable to traditional discrete networks. We introduce hypersolvers, neural networks designed to solve ODEs with low overhead and theoretical guarantees on accuracy. The synergistic combination of hypersolvers and Neural ODEs allows for cheap inference and unlocks a new frontier for practical application of continuous-depth models. Experimental evaluations on standard benchmarks, such as sampling for continuous normalizing flows, reveal consistent pareto efficiency over classical numerical methods.