  We investigate learning the eigenfunctions of evolution operators for
time-reversal invariant stochastic processes, a prime example being the
Langevin equation used in molecular dynamics. Many physical or chemical
processes described by this equation involve transitions between metastable
states separated by high potential barriers that can hardly be crossed during a
simulation. To overcome this bottleneck, data are collected via biased
simulations that explore the state space more rapidly. We propose a framework
for learning from biased simulations rooted in the infinitesimal generator of
the process and the associated resolvent operator. We contrast our approach to
more common ones based on the transfer operator, showing that it can provably
learn the spectral properties of the unbiased system from biased data. In
experiments, we highlight the advantages of our method over transfer operator
approaches and recent developments based on generator learning, demonstrating
its effectiveness in estimating eigenfunctions and eigenvalues. Importantly, we
show that even with datasets containing only a few relevant transitions due to
sub-optimal biasing, our approach recovers relevant information about the
transition mechanism.
