Fig 1.
(A) The selection of mathematical models. Images from various mathematical models are embedded in Contrastive Language-Image Pre-training (CLIP) latent space. Features of a target image are extracted, and the most similar pattern images are selected. (B) Parameter estimation. Three steps, feature extraction, dimensionality reduction, and Simulation-Decoupled Neural Posterior Estimation(SD-NPE), are conducted to predict the parameters corresponding to target images generated by a single parameter set.
Table 1.
Dataset for the selection of mathematical models.
Table 2.
Parameter set of Turing model for parameter estimation.
Fig 2.
The diagram of the modified contrastive learning for dimensionality reduction.
Parameter space is separated into regular grids to determine the positive and negative samples for the anchor. Only the triplets of generated images with easy negative are input to the multilayer perceptron (MLP) model, which is trained based on Triplet loss.
Table 3.
The configuration of the architecture for dimensionality reduction.
Table 4.
Training Conditions for Contrastive Learning.
Table 5.
Training Conditions for NGBoost.
Fig 3.
(A) Mapping of mathematical models in Contrastive Language-Image Pre-training (CLIP) latent space. The figure shows the results of visualizing CLIP’s 512-dimensional latent space, reduced to two dimensions using Uniform Manifold Approximation and Projection (UMAP). The horizontal and vertical axes represent the first and second components of the two-dimensional vectors obtained from UMAP, respectively. (B) The target image and the top three images with the highest similarity. (C) Representation of cosine similarity to the target Turing image. The figure shows the same scatter plot as in (A), with colors indicating the similarity to the target example. Star shows the position of the target image. The horizontal and vertical axes represent the first and second components of the two-dimensional vectors obtained from UMAP, respectively. (D) The histogram of cosine similarity scores of all datasets to the target image. (E) The definition of cosine similarity.
Fig 4.
Heatmap of MAP scores between patterns of mathematical models.
The heatmap displays the average Mean Average Precision (MAP) score for patterns that rank within the top 50 in similarity when a pattern image of a mathematical model is used as input. The vertical axis represents the type of mathematical model used as input, while the horizontal axis represents the type considered as the true label for MAP calculation. The range of possible MAP scores is [0, 1].
Fig 5.
Images of living organisms and high similarity patterns selected by CLIP.
Original images of living organisms (The leftmost column), target images after preprocessing (The second column from the left), and pattern images of top 5 similarities in the latent space of Contrastive Language-Image Pre-training (CLIP)(The third column from the left onwards). The target living organisms are emperor angelfish (Pomacanthus imperator), stripey (Microcanthus strigatus), areolate grouper (Epinephelus areolatus), humphead wrasse (Cheilinus undulatus), and mutants of Bacillus mycoides Flügge [18] from top to bottom.
Fig 6.
Parameter estimation of the Turing model.
(A) The process of Simulation-Decoupled Neural Posterior Estimation(SD-NPE). For prediction, each data sample is input into Natural Gradient Boosting (NGBoost) individually, which outputs a posterior distribution of parameters for that sample. By integrating these posterior distributions with precomputed approximate prior distributions, we can approximate the posterior distribution of parameters across all samples. (B) An example of the estimation of parameters fv and gv of the Turing model. Left: the approximated posterior distribution and the plots of points of target parameters and parameters with high and low probability. Right: Each line shows three examples of the Turing model images corresponding to the target, high, and low probability parameters, respectively.
Fig 7.
Turing mechanism and the analytical features.
(A) Relationship between analytical solution and predicted probability distribution. Probabilistic samplings of (fv, gv) (red) are mapped on the contour plot of analytically-obtained pattern characteristics (kmax and Dk. (B) Linear stability analysis of Turing pattern. The dispersion relation shows the relationship between the wavenumber k and the growth rate λ of its amplitude. kmax is the k when λ is at its maximum. Dk is calculated from kright and kleft, which are defined as the bounds on the range of k when λ is positive [20].
Fig 8.
Comparison between UMAP and the MLP trained with contrastive learning.
(A) Dimensional reduction by Uniform Manifold Approximation and Projection (UMAP). The horizontal and vertical axes represent the first and second components of the two-dimensional vectors obtained from UMAP, respectively. (B) Dimensional reduction by multilayer perceptron (MLP). The horizontal and vertical axes represent the first and second components of the two-dimensional output vectors of MLP, respectively. (C) The Turing pattern images of positions 1–6 in the feature space of UMAP (A) and that of MLP (B).
Fig 9.
Comparison of dimensional reduction of similar patterns.
(A) Distribution of patterns in Uniform Manifold Approximation and Projection (UMAP) feature space. (B) Distribution of patterns in contrastive learning feature space. (C-E) Examples of Turing patterns and predicted posterior distributions for (fv = 0.6735, gv = 0.9376) (Red points in (A-B)): (C) pattern examples, (D) UMAP-based dimensionality reduction, (E) contrastive learning-based dimensionality reduction. (F-H) Examples of Turing patterns and predicted posterior distributions for (fv = 0.650 and gv = 0.9817) (Blue points in (A-B)): (F) pattern examples, (G) UMAP-based dimensionality reduction, (H) contrastive learning-based dimensionality reduction.
Fig 10.
Comparison of the prediction errors with MLP model, UMAP, and without dimensionality reduction.
The horizontal axis represents the number of samples used for training. The vertical axis represents the value of the generalization error. Error bars indicate the maximum and minimum values among the five training sessions. The multilayer perceptron (MLP) model trained with contrastive learning (red) was more efficient than both Uniform Manifold Approximation and Projection (UMAP)(green) and the case without dimensionality reduction (blue), even when the sample size was large (indicated by the dashed box).