SlowGuess/NeurIPS_2023_OCR huggingface dataset:
MMGP for an output field. Let  $\mathcal{D} = \{(\mathbf{X}^i, \widehat{\mathbf{U}}_{k_0}^i)\}_{i=1}^n$  be a training dataset for one of the problems given by Equation (5b), i.e. for output field  $k_0$ . A multioutput GP is first trained to predict the output embeddings  $\widehat{\mathbf{U}}_{k_0}$ . The predictions of the GP are then decoded with the inverse PCA mapping, and morphed back to the original input mesh  $\mathcal{M}^i$ . Due to this last nonlinear operation, the posterior distribution of the output field of interest  $\mathbf{U}_{k_0}$  is no longer Gaussian. The predictive uncertainties are thus obtained through Monte Carlo simulations. This training procedure is repeated for each of the  $d$  output fields.
3.3 Properties of the methodology
The sequence of preprocessing operations, including mesh morphing, finite element interpolation, and PCA, leads to a non-linear dimensionality reduction. Leveraging these deterministic processes reduces the burden on the machine learning stage, potentially necessitating fewer training examples to achieve robust model performance on complex mesh-based data. In the numerical experiments presented in Section 4, the morphing technique is chosen a priori, with ongoing research focused on optimizing this morphing to minimize the number of PCA modes, which leads to a highly nonlinear dimensionality reduction stage that is finely tuned to the specific characteristics of the data.
Gaussian process regression between the input and output embeddings has several advantages. From a theoretical perspective, there exists conditions on the features of a continuous kernel so that it may approximate an arbitrary continuous target function [59]. Gaussian processes also come with built-in predictive uncertainties, that are marginally valid under the a priori Gaussian assumption. Nevertheless, the proposed methodology can be combined with any other regressor such as a deep neural network instead of the Gaussian process.
For clarity of the presentation, the MMGP methodology is illustrated with very simple, if not the simplest, morphing and dimensionality reduction techniques. Alternatives are possible for each algorithm brick. In particular, the fixed topology restriction may be lifted with other morphing algorithms, see Appendix B for more details.
Three regression problems are considered in order to assess the efficiency of the proposed methodology. The chosen datasets are first described in Section 4.1. The experimental setup is summarized in Section 4.2, and the results are discussed in Section 4.3.
Three datasets in computational fluid and solid mechanics are considered, described below, and summarized in Table 1. All the considered datasets involve geometric variabilities and meshes with possibly different number of nodes and edges. Additional details can also be found in Appendix A.
Table 1: Summary of the considered datasets with  $d_{\Omega}$ : dimension of the physical problem,  $p$ : number of input scalars,  $d$ : number of output fields,  $m$ : number of output scalars.
Datasets	train/test sizes	dΩ	p	d	m	Avg. # nodes
Rotor37	1000/200	3	2	2	4	29,773
Tensile2d	500/200	2	6	6	4	9,425
AirfRANS	800/200	2	2	3	2	179,779
AirfRANS-remeshed	800/200	2	2	3	2	19,527
Rotor37 dataset. We consider a 3D compressible steady-state Reynold-Averaged Navier-Stokes (RANS) simulation solved with elsA [18] using the finite volumes method. The inputs are given
by a mesh representing the surface of a 3D compressor blade [8], and two additional parameters that correspond to an input pressure and a rotation speed. The outputs of the problem are given by 4 scalars (massflow  $m$ , compression rate  $\tau$ , isentropic efficiency  $\eta$ , polyentropic efficiency  $\gamma$ ), and 2 fields on the surface of the input mesh (temperature  $T$ , pressure  $P$ ).
Tensile2d dataset. The second dataset corresponds to a 2D quasi-static problem in solid mechanics. The geometrical support consists of a 2D square, with two half-circles that have been cut off in a symmetrical manner. The inputs are given by a mesh, a pressure applied on the upper boundary, and 5 material parameters modeling the nonlinear elastoviscoplastic law of the material [50]. The boundary value problem is solved with the finite element method and the Z-set software [60]. The outputs of the problem are given by 4 scalars  $(p_{\mathrm{max}}, v_{\mathrm{max}}, \sigma_{22}^{\mathrm{max}}$ , and  $\sigma_v^{\mathrm{max}})$  and 6 fields of interest  $(u, v, p, \sigma_{11}, \sigma_{12}$ , and  $\sigma_{22}$ ). For the sake of brevity, the reader is referred to Appendix A for a description of these quantities.
AirfRANS dataset. The last dataset is made of 2D incompressible RANS around NACA profiles and taken from Bonnet et al. [14]. The inputs are given by a mesh of a NACA profile and two parameters that correspond to the inlet velocity and the angle of attack. The outputs are given by 2 scalars (drag  $C_D$  and lift  $C_L$  coefficients), and 3 fields (the two components of the fluid velocity  $u$  and  $v$ , and the pressure field  $p$ ). An additional version of this dataset is also considered, where the input meshes have been coarsened using the MMG remesher [1], and the output fields have been transferred to the coarsened meshes. The output scalars are unchanged. Illustrations of the input meshes can be found in the original paper [14].
Morphings. The Tutte's barycentric mapping onto the unit disk is used for morphing the meshes in the Rotor37 and Tensile2d datasets. The input meshes in the AirfRANS dataset are morphed onto the first mesh using RBF.
PCA embeddings. Embeddings of sizes 32 and 64 are retained for respectively the spatial coordinates and output fields in the Rotor37 and AirfRANS datasets. Smaller embeddings of sizes 8 are considered for both the spatial coordinates and output fields in the Tensile2d dataset. Note that for the Tensile2d and AirfRANS, a more effective variant of PCA has been used, which can easily deal with very large meshes (see, e.g., [24, 36]), with up to hundreds of millions of degrees of freedom. Details about this variant can be found in Appendix C.
Gaussian processes. Anisotropic Matern-5/2 kernels and zero mean functions are used for all the Gaussian processes priors. The lengthscales and nugget parameter are optimized by maximizing the marginal log-likelihood function with a L-BFGS algorithm and 10 random restarts using the GPy package [35].
Baselines. The performance of MMGP is compared with two baselines, namely, a graph convolutional neural network (GCNN) with a UNet-type architecture [29] and the GeneralConv [79], and MeshGraphNets (MGN) [63]. The hyperparameters of the GNNs are chosen by relying on a grid search. The GCNN and MGN models are implemented with PyTorch Geometric [25] and DGL [73], respectively. Additional details about the architectures and hyperparameters can be found in Appendix D. Due to the sizes of the input meshes in the AirfRANS dataset, the considered GNN-based baselines are prohibitively expensive. Similarly to the work of [28], the GNNs are trained using coarsened input meshes as described in Section 4.1. The output fields predicted on the coarse meshes are then transferred back on the original fine meshes thanks to finite element interpolation.
Evaluation metrics. Accuracy of the trained models is assessed by computing relative RMSE errors. Let  $\{\mathbf{U}_{\mathrm{ref}}^i\}_{i = 1}^{n_*}$  and  $\{\mathbf{U}_{\mathrm{pred}}^i\}_{i = 1}^{n_*}$  be respectively test observations and predictions of a given field of interest. The relative RMSE considered herein is defined as
$$
\mathrm {R R M S E} _ {f} (\mathbf {U} _ {\mathrm {r e f}}, \mathbf {U} _ {\mathrm {p r e d}}) = \left(\frac {1}{n _ {\star}} \sum_ {i = 1} ^ {n _ {\star}} \frac {\frac {1}{N ^ {i}} \| \mathbf {U} _ {\mathrm {r e f}} ^ {i} - \mathbf {U} _ {\mathrm {p r e d}} ^ {i} \| _ {2} ^ {2}}{\| \mathbf {U} _ {\mathrm {r e f}} ^ {i} \| _ {\infty} ^ {2}}\right) ^ {1 / 2},
$$
where it is recalled that  $N^i$  is the number of nodes in the mesh  $\mathcal{M}^i$ , and  $\max(\mathbf{U}_{\mathrm{ref}}^i)$  is the maximum entry in the vector  $\mathbf{U}_{\mathrm{ref}}^i$ . Similarly for scalar outputs, the following relative RMSE is computed:
$$
\mathrm {R R M S E} _ {s} (\mathbf {w} _ {\mathrm {r e f}}, \mathbf {w} _ {\mathrm {p r e d}}) = \left(\frac {1}{n _ {\star}} \sum_ {i = 1} ^ {n _ {\star}} \frac {| w _ {\mathrm {r e f}} ^ {i} - w _ {\mathrm {p r e d}} ^ {i} | ^ {2}}{| w _ {\mathrm {r e f}} ^ {i} | ^ {2}}\right) ^ {1 / 2}.
$$
Given that the input meshes may have different number of nodes, the coefficients of determination  $Q^2$  between the target and predicted output fields are computed by concatenating all the fields together. For each of the considered regression problems, training is repeated 10 times in order to provide uncertainties over the relative RMSE and  $Q^2$  scalar regression coefficients.
4.3 Results and discussion
Predictive performance. The relative RMSE and  $Q^2$  scalar regression coefficients are reported in Table 2 for all the considered experiments. While the GNN-based baselines achieve good performance, the MMGP model consistently outperforms them with lower errors. It is worth emphasizing that the field  $p$  and scalar  $p_{\mathrm{max}}$  of the Tensile2d dataset are particularly challenging, see Appendix E.2 for more details. They represent the accumulated plasticity in the mechanical piece, which are non-zero for a small fraction of the training set. Figure 2 shows the graphs of the output scalars predictions versus the targets on test set for the Tensile2d case. Figure 3 also shows examples of fields prediction in the case of the AirfRANS problem. The MMGP model is able to accurately reproduce the output fields, with relative errors mostly located near the tips of the airfoils.
Table 2: Means and standard deviations (gray) of the relative RMSE and  $Q^2$  scalar regression coefficients for all the considered datasets and quantities of interest (QoI) (best is bold).
RRMSE	Q2
QoI	GCNN	MGN	MMGP	GCNN	MGN	MMGP
Rotor37 dataset
m	4.4e-3 (5e-4)	5.4e-3 (7e-5)	5.0e-4 (3e-6)	0.9816 (4e-3)	0.9720 (5e-4)	0.9998 (3e-6)
p	4.4e-3 (5e-4)	5.3e-3 (7e-5)	4.8e-4 (1e-6)	0.9803 (5e-3)	0.9710 (9e-4)	0.9998 (2e-6)
η	3.1e-3 (7e-4)	7.2e-3 (7e-5)	5.0e-4 (3e-6)	0.9145 (4e-2)	0.5551 (2e-3)	0.9979 (1e-6)
γ	2.9e-3 (6e-4)	6.5e-3 (2e-5)	4.6e-4 (2e-7)	0.9068 (4e-2)	0.5257 (2e-3)	0.9977 (2e-6)
P	1.7e-2 (8e-4)	1.7e-2 (2e-3)	7.2e-3 (5e-4)	0.9863 (1e-3)	0.9866 (3e-3)	0.9973 (4e-4)
T	3.9e-3 (1e-4)	1.4e-2 (2e-3)	8.2e-4 (1e-5)	0.9930 (5e-4)	0.9956 (1e-3)	0.9997 (1e-5)
Tensile2d dataset
pmax	1.6e-0 (7e-1)	2.7e-1 (4e-2)	6.6e-1 (3e-1)	0.4310 (2e-1)	0.6400 (2e-1)	0.9435 (2e-2)
vmax	4.4e-2 (7e-3)	5.8e-2 (2e-2)	5.0e-3 (3e-5)	0.9245 (3e-2)	0.9830 (1e-2)	0.9999 (2e-5)
σ22	3.1e-3 (7e-4)	4.5e-3 (1e-3)	1.7e-3 (2e-5)	0.9975 (1e-3)	0.9958 (1e-3)	0.9993 (2e-5)
σmax	1.2e-1 (4e-2)	2.4e-2 (9e-3)	5.0e-3 (3e-5)	0.9723 (2e-2)	0.9801 (1e-2)	0.9997 (7e-6)
u	4.5e-2 (1e-2)	1.5e-2 (1e-3)	3.4e-3 (4e-5)	0.9623 (2e-2)	0.9270 (1e-2)	0.9997 (6e-6)
v	7.4e-2 (2e-2)	9.7e-2 (7e-3)	5.5e-3 (8e-5)	0.9559 (3e-2)	0.9322 (1e-2)	0.9995 (1e-5)
p	1.3e-1 (7e-2)	1.1e-1 (2e-2)	4.4e-2 (1e-2)	0.5691 (1e-1)	0.2626 (1e-1)	0.7785 (9e-2)
σ11	1.0e-1 (4e-2)	2.8e-2 (3e-3)	3.7e-3 (1e-4)	0.9304 (4e-2)	0.8693 (3e-2)	0.9999 (2e-6)
σ12	4.5e-2 (4e-3)	7.5e-3 (4e-4)	2.4e-3 (2e-5)	0.9617 (5e-3)	0.9868 (1e-3)	0.9999 (1e-6)
σ22	3.3e-2 (3e-3)	2.7e-2 (1e-3)	1.4e-3 (1e-5)	0.9662 (6e-3)	0.9782 (2e-3)	0.9999 (1e-6)
AirfRANS dataset
CD	6.1e-2 (2e-2)	4.9e-2 (7e-3)	3.3e-2 (2e-3)	0.9596 (2e-2)	0.9743 (1e-2)	0.9831 (2e-3)
CL	4.1e-1 (1e-1)	2.4e-1 (8e-2)	8.0e-3 (6e-4)	0.9776 (8e-3)	0.9851 (1e-2)	0.9999 (2e-6)
u	5.6e-2 (3e-3)	8.3e-2 (2e-3)	1.8e-2 (9e-5)	0.9659 (3e-3)	0.9110 (3e-3)	0.9749 (8e-5)
v	4.2e-2 (2e-3)	1.2e-1 (2e-3)	1.5e-2 (3e-5)	0.9683 (3e-3)	0.7516 (5e-3)	0.9806 (3e-5)
p	8.5e-2 (7e-3)	9.9e-2 (1e-2)	5.1e-2 (2e-5)	0.9602 (8e-3)	0.9390 (2e-2)	0.9934 (1e-5)
Uncertainty estimates. Once trained, the MMGP model provides access to predictive uncertainties for the output fields and scalars. Figure 4 shows an example of predicted pressure field for an
Figure 2: (Tensile2d) Test predictions versus test targets obtained for the output scalars of interest.
Figure 3: (AirfRANS) Test sample 787, fields of interest  $u$  ( $UX$ ),  $v$  ( $UY$ ) and  $p$ : (left) reference, (middle) MMGP prediction, (right) relative error.
arbitrary test input mesh of the Rotor37 experiment, together with the predictive variance and the point-wise relative absolute error. High relative errors are localized where the pressure field exhibits a discontinuity, known as a shock in compressor aerodynamics. The predictive variance is also higher near this region, reflecting that the GP-based surrogate model is uncertain about its prediction of the shock position. Figure 5 shows graphs of two output scalars with respect to the input pressure in the Tensile2d problem. The predictive intervals of the MMGP model are discriminative: they get wider as the input pressure falls out of the support of the training distribution. In order to assess the validity of the prediction intervals, we compute the prediction interval coverage probability (PICP), i.e. the average of test targets that fall into the  $95\%$  prediction interval. For the AirfRANS dataset, PICPs of  $93.05\%$  and  $93.5\%$  for respectively the outputs  $C_L$  and  $C_D$  are obtained by averaging the individual PICPs of 10 independent MMGP models. The prediction intervals are slightly over-confident but this could be corrected by e.g. conformalizing the Gaussian process [69].
Figure 4: (Rotor37) MMGP: prediction, predictive variance, and  $L^1$  relative error of the pressure field for an arbitrary geometry in the test dataset.
Figure 5: (Tensile2d) MMGP: graphs of the predicted  $v_{\mathrm{max}}$  and  $\sigma_v^{\mathrm{max}}$  with respect to the pressure, for four different test input meshes, and 11 values of input pressure that go beyond the training range (-50, -40), with 95% confidence intervals.
Computational times. The MMGP model can easily be trained on CPU hardware and with much lower computational times, see Table 3.
Table 3: Training computational times: GCNN and MGN on a Nvidia A100 Tensor Core GPU (neural network training), MMGP on a 48 cores Intel Xeon Gold 6342 CPU (Gaussian process regressors training). Between parenthesis are indicated the numbers of trainings carried-out to optimize hyperparameters (best is bold).
Dataset	GCNN	MGN	MMGP
Rotor37	(200 ×) 24 h	(6 ×) 13 h 14 min	(10 ×) 2 min 49 s
Tensile2d	(200 ×) 1 h 25 min	(6 ×) 6 h 50 min	(10 ×) 1 min 38 s
AirfRANS	(200 ×) 5 h 15 min	(6 ×) 5 h 00 min	(10 ×) 5 min 47 s
In summary, our proposed method presents an innovative approach to approximating field and scalar quantities of interest within the context of solving complex physics problems for design optimization. Our work introduces two key contributions: firstly, the utilization of mesh morphing pretreatment in conjunction with finite element interpolation, and secondly, the incorporation of shape embedding through dimensional reduction of coordinates, treating them as continuous fields over the geometric support. These innovations alleviate the machine learning task from the challenges of handling variable-sized samples and the need to learn implicit shape embedding. By reducing the dimensionality of inputs and outputs, our approach allows for the application of efficient Gaussian process regression. Notably, our MMGP model exhibits several key advantages. It can seamlessly handle very large meshes, is amenable to efficient CPU training, is fairly interpretable, demonstrates high accuracy in our experimental results, and provides readily available predictive uncertainties.
Future works will explore the extension of our method to accommodate time-dependent quantities of interest and investigate the optimization of the morphing process to enhance data compression and overall performance. Our research opens exciting avenues for advancing the capabilities of machine learning in the realm of physics-based design optimization.
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Available dataset and code The code corresponding to the two-dimensional solid mechanics case (Tensile2d) described in Section 4.1 is available at https://gitlab.com/drti/mmgp [3]. A documentation is available at https://mmgp.readthedocs.io/ [2], where details are provided on how to download the dataset Tensile2d and reproduce the corresponding numerical experiments.
Details regarding the datasets are provided in Appendix A. Morphing strategies and dimensionality reduction techniques are described in Appendices B and C. Details about the GNNs baselines are given in Appendix D. Finally, additional results about the considered experiments are gathered in Appendix E.
This section provides additional details regarding the synthetic datasets Tensile2d and Rotor37. Regarding the AirfRANS dataset, the reader is referred to [14].
A.1 Rotor37 dataset
Examples of input geometries are shown in Figure 6 together with the associated output pressure fields. While the geometrical variabilities are moderate, it can be seen that they have a significant impact on the output pressure field. A design of experiment for the input parameters of this problem are generated with maximum projection LHS method [43]. For each input mesh and set of input parameters, a three-dimensional aerodynamics problem is solved with RANS, as illustrated in, e.g. [8]. The output scalars of the problem are obtained by post-processing the three-dimensional velocity.
Figure 6: (Rotor37) Four geometries with their corresponding output pressure fields. The first panel shows the mesh, and the second to last panels show a superposition of the corresponding geometry and the mesh of the first one.
A.2 Tensile2d dataset
Examples of input geometries are shown in Figure 7. A two-dimensional boundary value problem in solid mechanics is considered, under the assumption of small perturbations (see, e.g. [12]). The partial differential equation is supplemented with Dirichlet and Neumann boundary conditions: the displacement on the lower boundary is fixed, while a uniform pressure is applied at the top. The input parameters of the problem are chosen to be the magnitude of the applied pressure and 5 parameters involved in the elasto-visco-plastic constitutive law of the material [50]. The outputs of the problem are chosen as the components of the displacement field,  $u$  and  $v$ , the entries of the Cauchy stress tensor,  $\sigma_{11}, \sigma_{22}, \sigma_{12}$ , and the cumulative plastic strain  $p$ . We also consider 4 output scalars obtained by post-processing the fields of interest: the maximum plastic strain  $p_{\mathrm{max}}$  across the geometry, the maximum vertical displacement  $v_{\mathrm{max}}$  at the top of the geometry, and the maximum normal stress  $\sigma_{22}^{\mathrm{max}}$  and Von Mises stress  $\sigma_v^{\mathrm{max}}$  across the geometry. It is worth emphasizing that the cumulative plastic strain  $p$  is challenging to predict, as illustrated in Section E.
Figure 7: (Tensile2d) Illustration of the four input meshes that are used in Figure 5.
B Morphing strategies
In this section, we briefly describe the Tutte's barycentric mapping [71] and the radial basis function morphing [9, 21] used in the considered experiments.
Tutte's barycentric mapping. For this method, we are limited to connected triangular surface meshes of fixed topology, either in a 2D or in a 3D ambient space. Tutte's barycentric mapping starts by setting the value of the displacement of the boundary points of the mesh (usually onto the unit disk), and solve for the value at all the remaining nodes of the mesh. The physical features available on the mesh, and inherited from the problem, are used in the specification of the displacement of the boundary nodes.