Learning function operators has gained significant attention in recent years, particularly in the context of solving parametric partial differential equations (PDEs). Prominent architectures include DeepONets [Lu21L] and Fourier neural operators (FNO) [Kov23N] with different properties, advantages or trade-offs.

An ideal operator learning framework should not require a particular choice of discretization for the input functions, allowing for the input and output functions to be on different domains, and being able to have different grids between samples.

A recent paper [Zha23B] outlines these challenges in operator learning nicely and introduces the following definitions of discretization-invariant, prediction-free, and domain-independent neural operators.

## Properties

**Discretization-invariant:**
Locations of sensors in the input function domain are not fixed.
This is important for unstructured input data, e.g., meshes.

**Prediction-free:**
Evaluation point can be any point in the output function domain.
This enables, e.g., physics-informed training or super-resolution.

**Domain-independent:**
Output function domain is independent of input function domain.
Provides much more flexibility, e.g., map boundary condition to solution.

Interestingly, the paper compares the properties of DeepONet and FNO, see Table 1, and shows that neither DeepONets nor FNOs satisfy all three properties.

Discretization-invariant | Prediction-free | Domain-independent | |
---|---|---|---|

DeepONet | 🚫 | ✅ | ✅ |

FNO | ✅ | 🚫 | 🚫 |

BelNet | ✅ | ✅ | ✅ |

**Table 1.**[Zha23B] Comparison of DeepONet, FNO and our BelNet.

## BelNet

The basis enhanced learning network (BelNet) [Zha23B], a new neural operator architecture proposed in the paper, is designed to meet these requirements. The derivation starts from the general neural operator framework [Kov23N], assumes a specific kernel structure, and uses quadrature rules to approximate the integral. In the end, the architecture introduces projection nets that project the input function into a latent space of learned basis functions.

BelNet is both a generalization of the networks proposed in [Che95U] (aka DeepONets [Lu21L]) and Fourier neural operators [Kov23N] by introducing a learnable set of basis functions for projection.

Through several challenging high-contrast and multi-scale problems, the paper shows that the approach outperforms other operator learning methods for these tasks and allows for more freedom in the sampling and discretization process.

Note that the universal approximation theorem for BelNet should follow from a generalization of [Che95U], but one needs to show that the approximation does not require fixed or similar grids.

## Conclusion

The paper is a great outline of independence of discretization in operator
learning, but particularly the idea of *learning* a projection of the
input function into a latent space is an interesting one, and it will be
exciting to see how this approach develops in the future.

If you are interested in trying out BelNet in a generalized framework, check out continuiti, our Python package for learning function operators with neural networks.