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Workshop: The Symbiosis of Deep Learning and Differential Equations

Enhancing the trainability and expressivity of deep MLPs with globally orthogonal initialization

Hanwen Wang · Paris Perdikaris


Multilayer Perceptrons (MLPs) defines a fundamental model class that forms the backbone of many modern deep learning architectures. Despite their universality guarantees, practical training via stochastic gradient descent often struggles to attain theoretical error bounds due to issues including (but not limited to) frequency bias, vanishing gradients, and stiff gradient flows. In this work we postulate that many of such issues find origins in the initialization of the network's parameters. While the initialization schemes proposed by Glorot {\it et al.} and He {\it et al.} have become the de-facto choices among practitioners, their goal to preserve the variance of forward- and backward-propagated signals is mainly achieved by assumptions on linearity, while the presence of nonlinear activation functions may partially destroy these efforts. Here, we revisit the initialization of MLPs from a dynamical systems viewpoint to explore why and how under these classical scheme, the MLP could still fail even at the beginning. Drawing inspiration from classical numerical methods for differential equations that leverage orthogonal feature representations, we propose a novel initialization scheme that promotes orthogonality in the features of the last hidden layer, ultimately leading to more diverse and localized features. Our results demonstrate that network initialization alone can be sufficient in mitigating frequency bias and yields competitive results for high-frequency function approximation and image regression tasks, without any additional modifications to the network architecture or activation functions.

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