Quantum computers have the potential to revolutionize diverse fields, including quantum chemistry, materials science, and machine learning. However, contemporary quantum computers experience errors that often cause quantum programs run on them to fail. Until quantum computers can reliably execute large quantum programs, stakeholders will need fast and reliable methods for assessing a quantum computer’s capability—i.e., the programs it can run and how well it can run them. Previously, off-the-shelf neural network architectures have been used to model quantum computers' capabilities, but with limited success, because these networks fail to learn the complex quantum physics that determines real quantum computers' errors. We address this shortcoming with a new quantum-physics-aware neural network architecture for learning capability models. Our scalable architecture combines aspects of graph neural networks with efficient approximations to the physics of errors in quantum programs. This approach achieves up to $\sim50\%$ reductions in mean absolute error on both experimental and simulated data, over state-of-the-art models based on convolutional neural networks, and scales to devices with 100+ qubits.