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Deep Neural Networks Motivated by Partial Differential Equations

Lars Ruthotto, Eldad Haber

arXiv:1804.04272v2cs.LGmath.OCstat.ML

TL;DR

Deep CNNs face architectural, computational, stability, and interpretability challenges. This paper interprets residual CNNs through PDEs, derives parabolic and hyperbolic architectures, and reports competitive image-classification performance in numerical experiments.

  • Problem

    Deep CNNs require difficult architecture choices and face substantial training costs, memory demands, stability issues, and limited interpretability.

  • Method

    The paper interprets residual CNNs as discretized differential equations and uses PDE theory to construct parabolic, first-order hyperbolic, and second-order hyperbolic CNNs.

  • Results

    The proposed architectures are reported as competitive in numerical image-classification experiments.

  • Takeaways & Limitations

    PDE analysis provides a framework for designing CNNs with distinct smoothing, energy, reversibility, and stability properties.

  • Takeaways & Limitations

    ResNet units keep feature dimensions fixed across layers, limiting practical applications and requiring additional dimension-changing layers.

Abstract

from arXiv · show

Partial differential equations (PDEs) are indispensable for modeling many physical phenomena and also commonly used for solving image processing tasks. In the latter area, PDE-based approaches interpret image data as discretizations of multivariate functions and the output of image processing algorithms as solutions to certain PDEs. Posing image processing problems in the infinite dimensional setting provides powerful tools for their analysis and solution. Over the last few decades, the reinterpretation of classical image processing problems through the PDE lens has been creating multiple celebrated approaches that benefit a vast area of tasks including image segmentation, denoising, registration, and reconstruction. In this paper, we establish a new PDE-interpretation of a class of deep convolutional neural networks (CNN) that are commonly used to learn from speech, image, and video data. Our interpretation includes convolution residual neural networks (ResNet), which are among the most promising approaches for tasks such as image classification having improved the state-of-the-art performance in prestigious benchmark challenges. Despite their recent successes, deep ResNets still face some critical challenges associated with their design, immense computational costs and memory requirements, and lack of understanding of their reasoning. Guided by well-established PDE theory, we derive three new ResNet architectures that fall into two new classes: parabolic and hyperbolic CNNs. We demonstrate how PDE theory can provide new insights and algorithms for deep learning and demonstrate the competitiveness of three new CNN architectures using numerical experiments.

1 Introduction

PDE-based modeling treats data as discretized functions, providing mathematical tools for analyzing processing tasks. The paper extends this perspective to deep CNNs and uses PDE theory to guide new architectures and understanding.

  • PDE-inspired methods model speech, image, and video data as discretized functions and operations as discretized PDE operators.This continuous formulation supports mathematical analysis and resolution-independent multiscale algorithms.
  • The paper establishes a PDE interpretation of deep learning tasks involving speech, images, and videos by continuously representing images and extending ODE-based analyses.
  • Deep CNN design requires choices about depth, width, layer operations, and filter connections, while training these properties jointly is computationally costly.
  • PDE theory provides mathematical guidance for CNN design by deriving architectures through suitable time discretizations of underlying PDEs.
  • The framework targets interpretability and robustness by placing learning within nonlinear PDE theory, while proposing parabolic and hyperbolic CNN architectures.The proposed architectures differ in smoothing, energy behavior, and potential memory requirements.
  • The paper presents three PDE-motivated CNN architectures and evaluates their competitiveness through numerical image-classification experiments.

2 Residual Networks and Differential Equations

The paper formulates residual CNNs as discretized differential equations and develops a PDE-based view of their layers, stability, convolutions, and resolution dependence. This perspective motivates new architectures and computational strategies.

  • Residual Networks and Differential Equations: CNN layers combine affine transformations, pointwise nonlinearities, normalization, and convolution operators to filter feature representations.For images, convolution operators are structured by channels and spatial filters; batch normalization uses channel statistics across spatial dimensions and examples.
  • Residual Networks and Differential Equations: A ResNet unit updates features through Yj+1 = Yj + F(θ(j), Yj), corresponding to a forward Euler discretization of an ordinary differential equation.Network depth corresponds to artificial integration time, linking learning to parameter estimation for nonlinear ODE systems.
  • Residual Networks and Differential Equations: ResNet feature dimensions remain fixed within a unit, limiting practical applications and requiring concatenation with layers that change resolution or channel count.
  • Residual Networks and Differential Equations: Training jointly estimates network and classifier parameters through a high-dimensional, non-convex optimization problem with costly per-example computation.Very deep architectures can also suffer vanishing or exploding gradients when forward propagation is unstable.
  • Convolutional ResNets and PDEs: A class of deep residual CNNs can be interpreted as nonlinear PDE systems by representing feature data as discretized continuous functions.The construction begins with one-dimensional convolutions and extends to higher dimensions and multiple channels.
  • Convolutional ResNets and PDEs: Resolution-dependent learned weights motivate coarse-to-fine multiscale training, reducing operations and memory during training and enabling coarsened evaluation.Coarse training may also reduce the risk of undesirable local minima by ignoring fine-scale features.

3 Deep Neural Networks motivated by PDEs

The paper restricts residual CNN architectures using PDE-inspired structure to obtain stability properties and improve memory efficiency. It introduces parabolic and hyperbolic designs whose discretizations support stable or reversible forward propagation under stated conditions.

  • Motivation: Standard CNN stability is difficult to guarantee because independently chosen convolution operators can yield unknown Jacobian spectral properties.This uncertainty complicates stability analysis and the selection of suitable time-integration methods.
  • Parabolic CNN: Symmetric convolutional layers produce negative semi-definite Jacobians for any non-decreasing activation function.This construction links the discrete network to PDE-inspired architectures and supports forward stability analysis.
  • Parabolic CNN: Parabolic CNN dynamics coincide with the heat equation for identity activation, identity normalization, and K(t) = ∇, connecting the model to image filtering PDEs.A total-variation-inspired normalization layer produces regular TV-denoising dynamics when the convolution reduces to a discrete gradient.
  • Parabolic CNN: For monotonically nondecreasing activations, parabolic CNN forward propagation is stable, and an appropriate Euler step size can preserve discrete stability.The discrete method requires δt to satisfy an eigenvalue-based stability condition; experiments limit convolution magnitudes through optimization constraints.
  • Hyperbolic CNNs: Hyperbolic CNNs exploit reversibility to avoid storing intermediate states, improving memory efficiency for very deep networks.Hamiltonian CNNs use symplectic Verlet integration, while second-order CNNs use Leapfrog discretization and are associated with nonlinear telegraph dynamics.
  • Hyperbolic CNNs: The second-order hyperbolic network is stable for stationary weights under an activation bound, with its energy bounded above by that of the linear wave equation.For time-dependent weights, weaker stability results are possible when the time derivative of the weights is bounded.

4 Regularization

The paper introduces temporal regularization to promote stable CNN dynamics and constrains weights to keep the time step sufficiently small.

  • A time-smoothness regularizer is proposed because stability requires the linear operator K not to change drastically over time.The paper contrasts an H1-seminorm with a piecewise-smooth function space for the kernels.
  • The regularizer combines a total-variation penalty that favors piecewise-constant dynamics with a smoothed ℓ1-norm term.The conditioning parameter satisfies τ > 0, and α1, α2 ≥ 0 are fixed regularization parameters.
  • The weights are constrained by −1 ≤ θ(1)(tj) ≤ 1 at every time point, while experiments fix the time step to δt = 1.The constraint is used because δt can be absorbed into K and must remain sufficiently small for stability.

5 Numerical Experiments

The experiments evaluate the proposed PDE-inspired CNNs on STL-10, CIFAR-10, and CIFAR-100 using a shared architecture, standard SGD training, and limited hyperparameter tuning.

  • Experimental setup: The proposed architectures are tested on STL-10, CIFAR-10, and CIFAR-100 to assess whether modeling restrictions still permit competitive classification.The experiments use random flipping and cropping for data augmentation.
  • Network Architecture: Each network uses an opening layer, ResNet blocks with time steps, and connectors that widen the CNN while coarsening the images.The experiments hold the opening and connecting layers fixed while varying the parabolic or hyperbolic ResNet block.
  • Network Architecture: The classifier consists of a fully connected layer, softmax transformation, and crossentropy loss, while the ResNet blocks use symmetric layers with total variation normalization and ReLU.The normalization parameter is ϵ = 10−3.
  • Training Algorithm: Training uses stochastic gradient descent with momentum 0.9 and piecewise-constant learning rates reduced at specified epochs.The epoch schedules differ between STL-10/CIFAR-10 and the more challenging CIFAR-100 dataset.
  • STL-10: STL-10 contains 13,000 96 × 96 color images across ten categories, split into 5,000 training and 8,000 test images.Its relatively small training set makes STL-10 a challenging image-classification benchmark.
  • STL-10: For STL-10, four ResNet blocks use 16, 32, 64, and 128 channels, producing 324,794 Hamiltonian-network weights versus 618,554 for the parabolic and second-order networks.All blocks use three time steps with δt = 1, total variation normalization, and ReLU activation.
  • STL-10: The data-availability experiment randomly increases the number of training examples, reserves 1,000 remaining examples for validation, and selects weights using validation accuracy.No data augmentation is used, and training accuracy is close to 100% in all cases.

B. Convergence for full data

On full-data benchmarks, the proposed architectures are competitive across datasets, with the second-order network slightly strongest on CIFAR-10 and CIFAR-100.

  • STL-10: The parabolic and first-order hyperbolic architectures outperform the second-order network in the STL-10 convergence example.The figure tracks test accuracy as training images increase from 10% to 80% and validation accuracy across SGD epochs.
  • STL-10: 77.0% and 78.3% vs. 74.3% test classification accuracy: the parabolic and Hamiltonian networks slightly outperform the second-order network on STL-10.The Hamiltonian network achieves the best test accuracy with about half as many trainable weights as the other two networks.
  • CIFAR-10 and CIFAR-100: CIFAR-10 and CIFAR-100 contain 60,000 labeled 32 × 32 RGB images each, with 50,000 used for training and validation and 10,000 for testing.CIFAR-10 has 10 categories, whereas CIFAR-100 has 100.
  • CIFAR-10 and CIFAR-100: The CIFAR networks use three parabolic or hyperbolic blocks while reducing image size from 32 × 32 to 8 × 8.
  • CIFAR-10 and CIFAR-100: The CIFAR architectures contain between 264,106 and 652,484 trainable weights, with more channels and a final connecting layer for CIFAR-100.
  • CIFAR-10 and CIFAR-100: The three architectures achieve comparable CIFAR results, while the second-order network slightly outperforms the others on these datasets.Additional tuning and more time steps produced about 5% higher CIFAR-10 accuracy and 9% higher CIFAR-100 accuracy in prior work.

6 Discussion and Outlook

The paper frames PDEs as a basis for stable, analyzable CNN design and reports competitive results despite architectural constraints, while leaving task-specific suitability unresolved.

  • 6 Discussion and Outlook: The PDE–CNN link provides a framework for designing, analyzing, and training residual CNNs, including resolution-dependent learned weights.The derived architectures are forward stable for parabolic dynamics and forward-backward stable for hyperbolic dynamics.
  • 6 Discussion and Outlook: Standard ResNet weights determine the underlying PDE type, making stability and numerical-method selection unknown before training.This motivates restricting CNNs a priori to stable nonlinear-PDE discretizations.
  • 6 Discussion and Outlook: The constrained architectures achieve adequate performance with relatively modest sizes, approaching state-of-the-art results from networks with considerably more weights.The authors caution that this may not hold generally and that future work must identify architectures suited to each learning task.
  • 6 Discussion and Outlook: PDE-based continuous CNN models are intended to streamline architecture design and improve training outcomes with less trial and error.
  • 6 Discussion and Outlook: Confusion matrices organize architectures by rows and increasing STL-10 training data by columns, counting predictions across 10 classes.Each matrix uses the full test set, with 800 images per class.
  • 6 Discussion and Outlook: Table 1 reports test classification accuracy, cross entropy loss, and trainable-weight counts after selecting weights with the best validation accuracy.Each experiment uses an 80% training and 20% validation split.

B. Convergence for CIFAR-100

Figure 4 compares validation accuracy for the three proposed architectures on CIFAR-10 and CIFAR-100 across stochastic-gradient-descent epochs. The architectures perform comparably, with the second-order network slightly ahead.

  • Figure 4 compares the three proposed architectures on CIFAR-10 and CIFAR-100.
  • Validation accuracy is shown at every stochastic-gradient-descent epoch using 10,000 randomly chosen images.
  • The architectures perform comparably, while the second-order network slightly outperforms the parabolic and first-order hyperbolic architectures.
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