How MEIDNet works

One latent space for two modalities

crystal (CIF) ──► SE(3)-equivariant encoder ──► z_c ─┐                     ┌─► crystal decoder   ──► lattice, species, positions
                                                      ├── shared space ─────┤
properties ────► property encoder (MLP) ─────► z_p ─┘  pulled together     └─► property decoder  ──► properties
                                                        (InfoNCE)

• The crystal encoder embeds each atom's element, passes messages along neighbour bonds using only squared distances (so rotating or translating the cell changes nothing), pools the atoms and adds the lattice.
• The property encoder maps the vector of (normalised) properties to the same space.
• Both are projected and normalised to unit length; a contrastive loss makes each structure's latent closest to its own property latent and far from the others in the batch. Its weight ramps up during training so the decoders stabilise first.
• The decoders are trained from the joint latent and from the property latent alone — that second term is what makes inverse design possible: a point chosen from properties must decode into a structure.

The training objective is

\[ L = w_1 L_\text{recon}(z_\text{joint}) + w_2 L_\text{prop}(z_\text{joint}) + w_3 L_\text{recon}(z_p) + w_4 L_\text{prop}(z_p) + \lambda(t)\, L_\text{InfoNCE}(z_c, z_p). \]

Inverse design = search in that space

For each target:

1. Start a population of latents at the property encoder's output for the target, plus a target-dependent random direction (so different targets start apart) and noise.
2. Optimise them by gradient descent on: distance of the decoded properties to the target (each objective with its loss and weight); a cosine anchor to the target latent; the family's search terms (atoms apart, cubic cell, positions near the prototype, expected charge zero, same element on equivalent sites, loose tolerance factor); and repulsion between population members and from earlier successes (diversity).
3. Decode: for each latent, choose one element per site group from the decoder's scores with temperature and top-k sampling; the last group is drawn only from elements whose charge balances the others. Place the elements on the prototype; the cell edge comes from ionic radii (a = 2(r_B + r_X) for perovskites).
4. Rules: symmetry-refine, then apply every hard constraint. Only compositions that pass all are kept.
5. Rank by distance to the target (select_weight, select_loss), skip duplicates and near-duplicate latents, save as CIF.

What "family" means

A family is data, not code: a prototype (site positions), site groups (allowed elements and oxidation states), a cell-size rule, soft search terms, and hard rules. The published perovskite tables are families/perovskite_abx3.yaml; the same machinery runs double_perovskite_a2bbx6.yaml and any file you write.

The design space

For a prototype family, the set of compositions the decoder can output is finite. meidnet space and the Studio enumerate it and score each composition with the structure encoder. The search's candidates are a subset of this space; comparing the two views (encoder prediction vs search's decoded prediction) is a useful sanity check that the reports encourage.

Reproducibility

MEIDNet 2 reproduces MEIDNet 1 exactly: the same featurisation, forward pass, training step and generation output (tests/test_v1_regression.py compares against the frozen v1 code in tests/legacy_v1/). One fix: v1's results depended on Python's hash seed through set iteration order; v2 fixes element order, so the same seed gives the same CIFs in any process.