Seismic data processing heavily relies on the solution of physics-driven inverse problems. In the presence of unfavourable data acquisition conditions (e.g., regular or irregular coarse sampling of sources and/or receivers), the underlying inverse problem becomes very ill-posed and prior information is required to obtain a satisfactory solution. Sparsity-promoting inversion, coupled with fixed-basis sparsifying transforms, represent the go-to approach for many processing tasks due to its simplicity of implementation and proven successful application in a variety of acquisition scenarios. Nevertheless, such transforms rely on the assumption that seismic data can be represented as a linear combination of a finite number of basis functions. Such an assumption may not always be fulfilled, thus producing sub-optimal solutions. Leveraging the ability of deep neural networks to find compact representations of complex, multi-dimensional vector spaces, we propose to train an AutoEncoder network to learn a nonlinear mapping between the input seismic data and a representative latent manifold. The trained decoder is subsequently used as a nonlinear preconditioner for the solution of the physics-driven inverse problem at hand. Through synthetic and field data examples, the proposed nonlinear, learned transformations are shown to outperform fixed-basis transforms and converge faster to the sought solution for a variety of seismic processing tasks, ranging from deghosting to wavefield separation with both regularly and irregularly subsampled data.