  We propose a new neural network based method for solving inverse problems for
partial differential equations (PDEs) by formulating the PDE inverse problem as
a bilevel optimization problem. At the upper level, we minimize the data loss
with respect to the PDE parameters. At the lower level, we train a neural
network to locally approximate the PDE solution operator in the neighborhood of
a given set of PDE parameters, which enables an accurate approximation of the
descent direction for the upper level optimization problem. The lower level
loss function includes the L2 norms of both the residual and its derivative
with respect to the PDE parameters. We apply gradient descent simultaneously on
both the upper and lower level optimization problems, leading to an effective
and fast algorithm. The method, which we refer to as BiLO (Bilevel Local
Operator learning), is also able to efficiently infer unknown functions in the
PDEs through the introduction of an auxiliary variable. Through extensive
experiments over multiple PDE systems, we demonstrate that our method enforces
strong PDE constraints, is robust to sparse and noisy data, and eliminates the
need to balance the residual and the data loss, which is inherent to the soft
PDE constraints in many existing methods.
