SymOnet: Expression-driven Foundation Model for Multi-operator Learning with Data-free Fine-tuning

Operator foundation models have advanced PDE solving through large-scale pre-training, enabling shared representations and transfer across physical systems. However, training and adapting these models typically require substantial high-fidelity solution data, which are expensive to obtain through numerical simulations or observations. To reduce this dependence during downstream adaptation, we introduce the Symbolic expression-driven Operator network foundation model (SymOnet) for data-free fine-tuning guided by governing equations, initial conditions, and boundary conditions. SymOnet learns a shared operator representation across heterogeneous PDE families through a token-query architecture that maps discretized input functions and space–time coordinates to solution values. Multi-PDE pre-training builds this representation, while downstream adaptation optimizes only lightweight adapters using expression-driven losses evaluated through automatic differentiation, keeping the shared backbone frozen. We establish a family-wise approximation theorem showing that a shared query basis can support operator approximation across a family of PDEs. At comparable model sizes, SymOnet outperforms all evaluated baselines on all 12 tasks under expression-driven fine-tuning, while also attaining the lowest L2RE on 9 tasks under data-driven fine-tuning. Additional experiments on three-dimensional compressible Navier–Stokes dynamics demonstrate transfer from two-dimensional pre-training to both fine-tuning regimes.