Then,itsamplesafixednumber of nodes by the calculated probability,and recursively conducts such procedure per layer toconstruct the whole computation graph.

Differential equations are widely used to describe complex dynamical systems with evolving parameters in nature and engineering. Effectively learning a family of maps from the parameter function to the system dynamics is of great significance. In this study, we propose a novel learning framework of symbolic continuous-depth neural networks, termed Symbolic Neural Ordinary Differential Equations (SNODEs), to effectively and accurately learn the underlying dynamics of complex systems. Specifically, our learning framework comprises three stages: initially, pre-training a predefined symbolic neural network via a gradient flow matching strategy; subsequently, fine-tuning this network using Neural ODEs; and finally, constructing a general neural network to capture residuals. In this process, we apply the SNODEs framework to partial differential equation systems through Fourier analysis, achieving resolution-invariant modeling. Moreover, this framework integrates the strengths of symbolism and connectionism, boasting a universal approximation theorem while significantly enhancing interpretability and extrapolation capabilities relative to state-of-the-art baseline methods. We demonstrate this through experiments on several representative complex systems. Therefore, our framework can be further applied to a wide range of scientific problems, such as system bifurcation and control, reconstruction and forecasting, as well as the discovery of new equations.

Learning from demonstration (LfD) provides an efficient way to train robots. The learned motions should be convergent and stable, but to be truly effective in the real world, LfD-capable robots should also be able to remember multiple motion skills. Existing stable-LfD approaches lack the capability of multi-skill retention. Although recent work on continual-LfD has shown that hypernetwork-generated neural ordinary differential equation solvers (NODE) can learn multiple LfD tasks sequentially, this approach lacks stability guarantees. We propose an approach for stable continual-LfD in which a hypernetwork generates two networks: a trajectory learning dynamics model, and a trajectory stabilizing Lyapunov function. The introduction of stability generates convergent trajectories, but more importantly it also greatly improves continual learning performance, especially in the size-efficient chunked hypernetworks. With our approach, a single hypernetwork learns stable trajectories of the robot's end-effector position and orientation simultaneously, and does so continually for a sequence of real-world LfD tasks without retraining on past demonstrations. We also propose stochastic hypernetwork regularization with a single randomly sampled regularization term, which reduces the cumulative training time cost for N tasks from O$(N^2)$ to O$(N)$ without any loss in performance on real-world tasks. We empirically evaluate our approach on the popular LASA dataset, on high-dimensional extensions of LASA (including up to 32 dimensions) to assess scalability, and on a novel extended robotic task dataset (RoboTasks9) to assess real-world performance. In trajectory error metrics, stability metrics and continual learning metrics our approach performs favorably, compared to other baselines. Our open-source code and datasets are available at https://github.com/sayantanauddy/clfd-snode.

In continual learning from demonstration (CLfD), a robot learns a sequence of real-world motion skills continually from human demonstrations. Recently, hypernetworks have been successful in solving this problem. In this paper, we perform an exploratory study of the effects of different optimizers, initializers, and network architectures on the continual learning performance of hypernetworks for CLfD. Our results show that adaptive learning rate optimizers work well, but initializers specially designed for hypernetworks offer no advantages for CLfD. We also show that hypernetworks that are capable of stable trajectory predictions are robust to different network architectures.