Operator learning for hyperbolic partial differential equations

We construct the first rigorously justified probabilistic algorithm for recovering the solution operator of a hyperbolic partial differential equation (PDE) in two variables from input-output training pairs. The primary challenge of recovering the solution operator of hyperbolic PDEs is the presence of characteristics, along which the associated Green's function is discontinuous. Therefore, a central component of our algorithm is a rank detection scheme that identifies the approximate location of the characteristics. By combining the randomized singular value decomposition with an adaptive hierarchical partition of the domain, we construct an approximant to the solution operator using $O(\Psi_\epsilon^{-1}\epsilon^{-7}\log(\Xi_\epsilon^{-1}\epsilon^{-1}))$ input-output pairs with relative error $O(\Xi_\epsilon^{-1}\epsilon)$ in the operator norm as $\epsilon\to0$, with high probability. Here, $\Psi_\epsilon$ represents the existence of degenerate singular values of the solution operator, and $\Xi_\epsilon$ measures the quality of the training data. Our assumptions on the regularity of the coefficients of the hyperbolic PDE are relatively weak given that hyperbolic PDEs do not have the ``instantaneous smoothing effect'' of elliptic and parabolic PDEs, and our recovery rate improves as the regularity of the coefficients increases.

BrainBERT: Self-supervised representation learning for intracranial recordings

We create a reusable Transformer, BrainBERT, for intracranial recordings bringing modern representation learning approaches to neuroscience. Much like in NLP and speech recognition, this Transformer enables classifying complex concepts, i.e., decoding neural data, with higher accuracy and with much less data by being pretrained in an unsupervised manner on a large corpus of unannotated neural recordings. Our approach generalizes to new subjects with electrodes in new positions and to unrelated tasks showing that the representations robustly disentangle the neural signal. Just like in NLP where one can study language by investigating what a language model learns, this approach opens the door to investigating the brain by what a model of the brain learns. As a first step along this path, we demonstrate a new analysis of the intrinsic dimensionality of the computations in different areas of the brain. To construct these representations, we combine a technique for producing super-resolution spectrograms of neural data with an approach designed for generating contextual representations of audio by masking. In the future, far more concepts will be decodable from neural recordings by using representation learning, potentially unlocking the brain like language models unlocked language.

Learning a natural-language to LTL executable semantic parser for grounded robotics

Children acquire their native language with apparent ease by observing how language is used in context and attempting to use it themselves. They do so without laborious annotations, negative examples, or even direct corrections. We take a step toward robots that can do the same by training a grounded semantic parser, which discovers latent linguistic representations that can be used for the execution of natural-language commands. In particular, we focus on the difficult domain of commands with a temporal aspect, whose semantics we capture with Linear Temporal Logic, LTL. Our parser is trained with pairs of sentences and executions as well as an executor. At training time, the parser hypothesizes a meaning representation for the input as a formula in LTL. Three competing pressures allow the parser to discover meaning from language. First, any hypothesized meaning for a sentence must be permissive enough to reflect all the annotated execution trajectories. Second, the executor -- a pretrained end-to-end LTL planner -- must find that the observed trajectories are likely executions of the meaning. Finally, a generator, which reconstructs the original input, encourages the model to find representations that conserve knowledge about the command. Together these ensure that the meaning is neither too general nor too specific. Our model generalizes well, being able to parse and execute both machine-generated and human-generated commands, with near-equal accuracy, despite the fact that the human-generated sentences are much more varied and complex with an open lexicon. The approach presented here is not specific to LTL; it can be applied to any domain where sentence meanings can be hypothesized and an executor can verify these meanings, thus opening the door to many applications for robotic agents.