A Neural ODE Interpretation of Transformer Layers

Transformer layers, which use an alternating pattern of multi-head attention and multi-layer perceptron (MLP) layers, provide an effective tool for a variety of machine learning problems. As the transformer layers use residual connections to avoid the problem of vanishing gradients, they can be viewed as the numerical integration of a differential equation. In this extended abstract, we build upon this connection and propose a modification of the internal architecture of a transformer layer. The proposed model places the multi-head attention sublayer and the MLP sublayer parallel to each other. Our experiments show that this simple modification improves the performance of transformer networks in multiple tasks. Moreover, for the image classification task, we show that using neural ODE solvers with a sophisticated integration scheme further improves performance

STEER: Simple Temporal Regularization For Neural ODEs

Training Neural Ordinary Differential Equations (ODEs) is often computationally expensive. Indeed, computing the forward pass of such models involves solving an ODE which can become arbitrarily complex during training. Recent works have shown that regularizing the dynamics of the ODE can partially alleviate this. In this paper we propose a new regularization technique: randomly sampling the end time of the ODE during training. The proposed regularization is simple to implement, has negligible overhead and is effective across a wide variety of tasks. Further, the technique is orthogonal to several other methods proposed to regularize the dynamics of ODEs and as such can be used in conjunction with them. We show through experiments on normalizing flows, time series models and image recognition that the proposed regularization can significantly decrease training time and even improve performance over baseline models.Comment: Neurips 202

PAC bounds of continuous Linear Parameter-Varying systems related to neural ODEs

We consider the problem of learning Neural Ordinary Differential Equations (neural ODEs) within the context of Linear Parameter-Varying (LPV) systems in continuous-time. LPV systems contain bilinear systems which are known to be universal approximators for non-linear systems. Moreover, a large class of neural ODEs can be embedded into LPV systems. As our main contribution we provide Probably Approximately Correct (PAC) bounds under stability for LPV systems related to neural ODEs. The resulting bounds have the advantage that they do not depend on the integration interval.Comment: 12 page

"Hey, that's not an ODE": Faster ODE Adjoints with 12 Lines of Code

Neural differential equations may be trained by backpropagating gradients via the adjoint method, which is another differential equation typically solved using an adaptive-step-size numerical differential equation solver. A proposed step is accepted if its error, \emph{relative to some norm}, is sufficiently small; else it is rejected, the step is shrunk, and the process is repeated. Here, we demonstrate that the particular structure of the adjoint equations makes the usual choices of norm (such as $L^2$) unnecessarily stringent. By replacing it with a more appropriate (semi)norm, fewer steps are unnecessarily rejected and the backpropagation is made faster. This requires only minor code modifications. Experiments on a wide range of tasks---including time series, generative modeling, and physical control---demonstrate a median improvement of 40% fewer function evaluations. On some problems we see as much as 62% fewer function evaluations, so that the overall training time is roughly halved

Characteristic Neural Ordinary Differential Equations

We propose Characteristic-Neural Ordinary Differential Equations (C-NODEs), a framework for extending Neural Ordinary Differential Equations (NODEs) beyond ODEs. While NODEs model the evolution of a latent variables as the solution to an ODE, C-NODE models the evolution of the latent variables as the solution of a family of first-order quasi-linear partial differential equations (PDEs) along curves on which the PDEs reduce to ODEs, referred to as characteristic curves. This in turn allows the application of the standard frameworks for solving ODEs, namely the adjoint method. Learning optimal characteristic curves for given tasks improves the performance and computational efficiency, compared to state of the art NODE models. We prove that the C-NODE framework extends the classical NODE on classification tasks by demonstrating explicit C-NODE representable functions not expressible by NODEs. Additionally, we present C-NODE-based continuous normalizing flows, which describe the density evolution of latent variables along multiple dimensions. Empirical results demonstrate the improvements provided by the proposed method for classification and density estimation on CIFAR-10, SVHN, and MNIST datasets under a similar computational budget as the existing NODE methods. The results also provide empirical evidence that the learned curves improve the efficiency of the system through a lower number of parameters and function evaluations compared with baselines