Fig. 4: Cost functions and their effects.

Inferring the spin Hamiltonian for Dy₂Ti₂O₇. a \(\chi _{S({\mathbf{Q}})}^2\) directly measures the distance between experimental and simulated S(Q) data. It is relatively flat and noisy around its minimum, thus yielding a large uncertainty in the J₂ coupling (any value below the dashed line, \(C_{S\left( {\mathbf{Q}} \right)}^2\), is a reasonable candidate). b \(\chi _{S_{{\mathrm{AE}}}({\mathbf{Q}})}^2\) measures the distance between S(Q) data after being filtered through the autoencoder. c \(\chi _{S_L}^2\) measures the distance between the 30-dimensional latent space representations of the S(Q) data. The Gaussian process model \(\hat \chi _{S_L}^2\left( H \right)\)(red curve) accurately approximates \(\chi _{S_L}^2\), even in the full space of J₂, J₃, and \(J_{3^\prime}\). d Once a good model \(\hat \chi _{S_L}^2(H)\) has been constructed, we can rapidly identify the optimal Hamiltonian model (magenta cross). The autoencoder-based error measure \(\chi _{S_L}^2\) yields much smaller model uncertainty (blue region) than the naïve one \(\chi _{S({\mathbf{Q}})}^2\) (cyan region). Model uncertainty is further reduced using a multi-objective error measure \(\chi _{{\mathrm{multi}}}^2\) that incorporates heat capacity data (dark-blue region). Three most popular Hamiltonian sets currently used from refs. ^17,18,20 have also been marked as green, red and gray crosses respectively.