Circuits in the brain commonly exhibit modular architectures that factorise complex tasks, resulting in the ability to compositionally generalise and reduce catastrophic forgetting. In contrast, artificial neural networks (ANNs) appear to mix all processing, because modular solutions are difficult to find as they are vanishing subspaces in the space of possible solutions. Here, we draw inspiration from fault-tolerant computation and the Poisson-like firing of real neurons to show that activity-dependent neural noise, combined with nonlinear neural responses, drives the emergence of solutions that reflect an accurate understanding of modular tasks, corresponding to acquisition of a correct world model. We find that noise-driven modularisation can be recapitulated by a deterministic regulariser that multiplicatively combines weights and activations, revealing rich phenomenology not captured in linear networks or by standard regularisation methods. Though the emergence of modular structure requires sufficiently many training samples (exponential in the number of modular task dimensions), we show that pre-modularised ANNs exhibit superior noise-robustness and the ability to generalise and extrapolate well beyond training data, compared to ANNs without such inductive biases. Together, our work demonstrates a regulariser and architectures that could encourage modularity emergence to yield functional benefits.