Here, we present novel adaptations for convolutional neural networks to demonstrate that they are indeed able to process functions as inputs and outputs.

Another disadvantage of these methods lies in the unavoidable resolving stage when the initial conditions of PDEs are varied.

Here, we present novel adaptations for convolutional neural networks to demonstrate that they are indeed able to process functions as inputs and outputs.

We obtain this result by discarding the high-frequency components, e.g. higher than frequency This proves the statement of the lemma. Appendix A.2 can be adapted to bandlimited Note that the single-channel versions were defined in the main text. We will describe these filters later in the text (see C.1.4) We use the same notation as in Section 2 and Appendix A.2. Hence, they consist of three elementary mappings between spaces of bandlimited functions, i.e., We recall that the convolutional operator appearing in (A.8) takes the form K As in Appendix A.2, the above proofs can be readily adapted We present the proof of a generalization of the universality result of Theorem 3.1. In addition, we will use the following notation in the proof.

Summary and Contributions: This paper provides an investigation of out-of-distribution generalization in visual question answering, as benchmarked by prior works on the VQA-CP dataset. The VQA-CP dataset by Agrawal et al. has different distributions in training and test, intentionally constructed so to encourage models to truly perform reasoning and generalize better, instead of naively picking up on question-only biases in the dataset. However, the authors demonstrate how several prior works on VQA-CP have (inadvertently) gamed this evaluation dataset without necessarily making progress due to a number of issues -- 1) exploiting knowledge of how the train/test splits were constructed to build models such that a) models are conditioned on the question prefix (and so will only work well on VQA-CP test and not generalize beyond), or b) poorly fit the training set. Next, the authors provide a few naive baselines that exploit the aforementioned issues (and as the authors acknowledge -- is not useful for any practical purposes) and perform well on VQA-CP test -- 1) a random predictions model that inverts the predicted answer distribution from training to test, and 2) a learned BUTD model that artificially ignores the top-predicted answer on VQA-CP test. The fact that a random predictions inverted model performs better on number and yes/no questions -- the question set that constitutes the largest fraction of performance -- is alarming, and provides a necessary and timely check on prior works on VQA-CP.

Out-of-distribution (OOD) testing is increasingly popular for evaluating a machine learning system's ability to generalize beyond the biases of a training set. OOD benchmarks are designed to present a different joint distribution of data and labels between training and test time. VQA-CP has become the standard OOD benchmark for visual question answering, but we discovered three troubling practices in its current use. First, most published methods rely on explicit knowledge of the construction of the OOD splits. They often rely on inverting'' the distribution of labels, e.g.

Although very successfully used in conventional machine learning, convolution based neural network architectures -- believed to be inconsistent in function space -- have been largely ignored in the context of learning solution operators of PDEs. Here, we present novel adaptations for convolutional neural networks to demonstrate that they are indeed able to process functions as inputs and outputs. The resulting architecture, termed as convolutional neural operators (CNOs), is designed specifically to preserve its underlying continuous nature, even when implemented in a discretized form on a computer. We prove a universality theorem to show that CNOs can approximate operators arising in PDEs to desired accuracy. CNOs are tested on a novel suite of benchmarks, encompassing a diverse set of PDEs with possibly multi-scale solutions and are observed to significantly outperform baselines, paving the way for an alternative framework for robust and accurate operator learning. Our code is publicly available at https://github.com/bogdanraonic3/ConvolutionalNeuralOperator