In the control problems of the PDE systems, observation is important to make the decision. However, the observation is generally sparse and missing in practice due to the limitation and fault of sensors. The above challenges cause observations with uncertain quantities and modalities. Therefore, how to leverage the uncertain observations as the states in control problems of the PDE systems has become a scientific problem. The dynamics of PDE systems rely on the initial conditions, boundary conditions, and PDE formula. Given the above three elements, PINNs can be used to solve the PDE systems. In this work, we discover that the neural network can also be used to identify and represent the PDE systems using PDE loss and sparse data loss. Inspired by the above discovery, we propose a Physics-Informed Representation (PIR) algorithm for multimodal policies in PDE systems' control. It leverages PDE loss to fit the neural network and data loss calculated on the observations with random quantities and modalities to propagate the information of initial conditions and boundary conditions into the inputs. The inputs can be the learnable parameters or the output of the encoders. Then, under the environments of the PDE systems, such inputs are the representation of the current state. In our experiments, the PIR illustrates the superior consistency with the features of the ground truth compared with baselines, even when there are missing modalities. Furthermore, PIR has been successfully applied in the downstream control tasks where the robot leverages the learned state by PIR faster and more accurately, passing through the complex vortex street from a random starting location to reach a random target.
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