Generic bounds on the approximation error for physics-informed (and) operator learning
This work provides foundational theoretical guarantees for physics-informed machine learning methods, addressing a key challenge in applying neural networks to PDEs, though it is incremental as it builds on existing approximation results.
The authors tackled the problem of deriving rigorous approximation error bounds for physics-informed neural networks (PINNs) and operator learning architectures like DeepONets and FNOs, showing that these methods can efficiently approximate solutions or solution operators of generic PDEs and mitigate the curse of dimensionality for nonlinear parabolic PDEs.
We propose a very general framework for deriving rigorous bounds on the approximation error for physics-informed neural networks (PINNs) and operator learning architectures such as DeepONets and FNOs as well as for physics-informed operator learning. These bounds guarantee that PINNs and (physics-informed) DeepONets or FNOs will efficiently approximate the underlying solution or solution operator of generic partial differential equations (PDEs). Our framework utilizes existing neural network approximation results to obtain bounds on more involved learning architectures for PDEs. We illustrate the general framework by deriving the first rigorous bounds on the approximation error of physics-informed operator learning and by showing that PINNs (and physics-informed DeepONets and FNOs) mitigate the curse of dimensionality in approximating nonlinear parabolic PDEs.