Neural Networks Trained to Solve Differential Equations Learn General Representations
We introduce a technique based on the singular vector canonical correlation analysis (SVCCA) for measuring the generality of neural network layers across a continuously-parametrized set of tasks. We illustrate this method by studying generality in neural networks trained to solve parametrized boundary value problems based on the Poisson partial differential equation. We find that the first hidden layers are general, and that they learn generalized coordinates over the input domain. Deeper layers are successively more specific. Next, we validate our method against an existing technique that measures layer generality using transfer learning experiments. We find excellent agreement between the two methods, and note that our method is much faster, particularly for continuously-parametrized problems. Finally, we also apply our method to networks trained on MNIST, and show it is consistent with, and complimentary to, another study of intrinsic dimensionality.
SUMMARY The paper presents a series of experiments in which singular vector canonical correlation analysis (SVCCA) is used to measure the similarity between neural network representations learned from different random initializations and from slightly different problems. In most prior work, the generalizability of neural network representations is measured through transfer learning experiments, which can be computationally expensive; the key insight of this paper is that SVCCA provides an alternative way to measure feature generalizability that tends to give similar conclusions as transfer learning, but which is orders of magnitude faster to compute.  The authors use SVCCA to perform extensive experiments measuring the degree to which differential equation neural networks (DENN) vary with different random seeds, and vary across related problem instances. Experiments with SVCCA suggest that lower network layers are more general, while higher layers are more specific to the particular problem instance on which the network was trained. Lower layers are more reproducible, giving rise to representations that are more stable under retraining, and across all layers reproducibility grows as network width increases. A subset of these experiments are reproduced with transfer learning, and give similar conclusions but take orders of magnitude longer to run.  PROS - The paper demonstrates that SVCCA can be a computationally inexpensive way to study neural network representations - Experiments with SVCCA suggest a number of interesting features of neural networks - The paper is extremely well-written and clearly presented  CONS - The motivation for focusing on DENNs is unclear - I don’t know how much the conclusions drawn from experiments on DENNs would generalize to other more commonly-used applications of neural networks.  WHY FOCUS ON DENNS? The paper motivates the use of DENNs with two major reasons: (Lines 61-68): First, that DENNs are a powerful method for solving differential equations, and second that they “offer the opportunity to study the behavior of neural networks in a well-understood context”. I agree that these are both interesting reasons for studying DENNs; however as presented the bulk of the paper has very little to do with DENNs at all, and instead focuses on the degree to which neural network representations vary when trained in different conditions. The questions that the paper raises around the intrinsic dimensionality, reproducibility, and specificity of learned representations are very interesting and of extremely broad applicability; however given the far-reaching implications of these experimental questions, I’m surprised that the authors chose to focus on the relatively niche area of DENNs rather than more well-studied applications like image recognition, speech recognition, language modeling, or machine translation.  HOW MUCH DO CONCLUSIONS GENERALIZE? The paper’s experiments suggest a number of intriguing conclusions about learned representations; however due to the focus on DENNs I’m unsure to what extent these conclusions would generalize to more standard applications of neural networks, especially since there are some quirks of DENNs that are quite different from many other types of neural networks.  First, the problem setup and loss for DENNs are not very representative of typical applications of neural networks; the network inputs the (x, y) coordinates of a point the plane, and outputs a scalar value; the loss function (Supplementary Equation 1) incorporates a squared Euclidean distance between the second derivative of the network’s output and a target function s. This problem setup and loss function are extremely different from most applications of neural networks, which typically input some natural data (images, text, speech, etc) and are trained with cross-entropy or regression loss functions.  Second, all networks studied are extremely small: they are four-layer fully-connected layers, with between 8 and 192 hidden units per layer. This means that the largest networks studied have on the order of 150K learnable parameters; in comparison a ResNet-50 on ImageNet has about 25M learnable parameters. Will the conclusions of the paper generalize to larger networks?  Third, all networks use tanh nonlinearities; ReLU nonlinearites are much more common for most problems. The authors do provide a convincing explanation for why tanh is preferred over ReLU for DENNs (Supplementary Lines 10-16), though I still wonder whether the results would apply to ReLU networks.  OVERALL On the whole I enjoyed reading the paper. It was extremely well-written, and used well-designed experiments and metrics to draw several interesting conclusions about neural network representations, demonstrating that SVCCA can be a valuable tool in understanding neural networks. However, due to the exclusive focus on DENNs I am unsure how much the conclusions will apply to other more widely-used types of neural networks; experiments on standard tasks (image classification, language modeling, etc) would have significantly strengthened the paper. Despite this shortcoming, overall I lean toward acceptance.  FINAL SCORE I appreciate the additional experiments presented on ReLU networks on MNIST, and I hope these can be included in the camera-ready version of the paper.