Recent work in equivariant deep learning bears strong similarities to physics. Fields over a base space are fundamental entities in both subjects, as are equivariant maps between these fields. In deep learning, however, these maps are usually defined by convolutions with a kernel, whereas they are partial differential operators (PDOs) in physics. Developing the theory of equivariant PDOs in the context of deep learning could bring these subjects even closer together and lead to a stronger flow of ideas. In this work, we derive a G-steerability constraint that completely characterizes when a PDO between feature vector fields is equivariant, for arbitrary symmetry groups G. We then fully solve this constraint for several important groups. We use our solutions as equivariant drop-in replacements for convolutional layers and benchmark them in that role. Finally, we develop a framework for equivariant maps based on Schwartz distributions that unifies classical convolutions and differential operators and gives insight about the relation between the two. However, one remaining difference is that physics uses equivariant partial differential operators (PDOs) to define maps between fields, such as the gradient or Laplacian. Therefore, using PDOs instead of convolutions in deep learning would complete the analogy to physics and could lead to even more transfer of ideas between subjects.Equivariant PDO-based networks have already been designed in prior work [21][22][23]. Most relevant for our work are PDO-eConvs [22], which can be seen as the PDO-analogon of group convolutions. * Work done during an internship at QUVA Lab Preprint. Under review.

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In many real-world applications, the reward function is too complex to be manually specified. In such cases, reward functions must instead be learned from human feedback. Since the learned reward may fail to represent user preferences, it is important to be able to validate the learned reward function prior to deployment. One promising approach is to apply interpretability tools to the reward function to spot potential deviations from the user's intention. Existing work has applied general-purpose interpretability tools to understand learned reward functions. We propose exploiting the intrinsic structure of reward functions by first preprocessing them into simpler but equivalent reward functions, which are then visualized. We introduce a general framework for such reward preprocessing and propose concrete preprocessing algorithms. Our empirical evaluation shows that preprocessed rewards are often significantly easier to understand than the original reward.

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