  Many systems may switch to an undesired state due to internal failures or
external perturbations, of which critical transitions toward degraded ecosystem
states are a prominent example. Resilience restoration focuses on the ability
of spatially-extended systems and the required time to recover to their desired
states under stochastic environmental conditions. While mean-field approaches
may guide recovery strategies by indicating the conditions needed to
destabilize undesired states, these approaches are not accurately capturing the
transition process toward the desired state of spatially-extended systems in
stochastic environments. The difficulty is rooted in the lack of mathematical
tools to analyze systems with high dimensionality, nonlinearity, and stochastic
effects. We bridge this gap by developing new mathematical tools that employ
nucleation theory in spatially-embedded systems to advance resilience
restoration. We examine our approach on systems following mutualistic dynamics
and diffusion models, finding that systems may exhibit single-cluster or
multi-cluster phases depending on their sizes and noise strengths, and also
construct a new scaling law governing the restoration time for arbitrary system
size and noise strength in two-dimensional systems. This approach is not
limited to ecosystems and has applications in various dynamical systems, from
biology to infrastructural systems.
