Bayesian Optimization in Reduced Eigenbases David Gaudrie 1 , - PowerPoint PPT Presentation

Parametric shape optimization Intrinsic dimensionality Meta-modeling in eigenbasis Optimization in eigenbasis Conclusions Bayesian Optimization in Reduced Eigenbases David Gaudrie 1 , Rodolphe Le Riche 2 , Victor Picheny 3 , ıt Enaux 1 , Vincent Herbert 1 Benoˆ 1Groupe PSA 2CNRS LIMOS, ´ erieure des Mines de Saint-´ Ecole Nationale Sup´ Etienne 3 Prowler.io PGMO Days, Saclay, December 4 th 2019

Parametric shape optimization Intrinsic dimensionality Meta-modeling in eigenbasis Optimization in eigenbasis Conclusions Parametric shape optimization Industrial context: Groupe PSA aims to optimize systems such as vehicle shape, combustion chamber design, ...

Parametric shape optimization Intrinsic dimensionality Meta-modeling in eigenbasis Optimization in eigenbasis Conclusions Parametric shape optimization Industrial context: Groupe PSA aims to optimize systems such as vehicle shape, combustion chamber design, ... min x ∈ X f ( x ), where x are CAD parameters of a shape Ω x .

Parametric shape optimization Intrinsic dimensionality Meta-modeling in eigenbasis Optimization in eigenbasis Conclusions Parametric shape optimization Industrial context: Groupe PSA aims to optimize systems such as vehicle shape, combustion chamber design, ... min x ∈ X f ( x ), where x are CAD parameters of a shape Ω x . NACA 22 airfoil Electric machine rotor

Parametric shape optimization Intrinsic dimensionality Meta-modeling in eigenbasis Optimization in eigenbasis Conclusions Parametric shape optimization Industrial context: Groupe PSA aims to optimize systems such as vehicle shape, combustion chamber design, ... min x ∈ X f ( x ), where x are CAD parameters of a shape Ω x Other CAD parameters → other shapes NACA 22 airfoil Electric machine rotor

Parametric shape optimization Intrinsic dimensionality Meta-modeling in eigenbasis Optimization in eigenbasis Conclusions Parametric shape optimization Expensive computer code Restricted number of evaluations ( ≈ 100-200), Limited knowledge of f ( · ) ⇒ use of surrogate models (GP) ⇒ Bayesian optimization.

Parametric shape optimization Intrinsic dimensionality Meta-modeling in eigenbasis Optimization in eigenbasis Conclusions Parametric shape optimization Expensive computer code Restricted number of evaluations ( ≈ 100-200), Limited knowledge of f ( · ) ⇒ use of surrogate models (GP) ⇒ Bayesian optimization. Curse of dimensionality: x ∈ X ⊂ R d , d � 50 ⇒ too many design variables with regard to the available budget. Computation time of a shape Ω x negligible compared to the evaluation of f ( x ). Assumption: the shapes lie in a δ < d dimensional manifold.

Parametric shape optimization Intrinsic dimensionality Meta-modeling in eigenbasis Optimization in eigenbasis Conclusions Parametric shape optimization Expensive computer code Restricted number of evaluations ( ≈ 100-200), Limited knowledge of f ( · ) ⇒ use of surrogate models (GP). Curse of dimensionality: x ∈ X ⊂ R d , d � 50 ⇒ too many design variables with regard to the available budget. Computation time of a shape Ω x negligible compared to the evaluation of f ( x ). Assumption: the shapes lie in a δ < d dimensional manifold. ⇒ Discover the δ -dimensional manifold, do kriging, and perform the optimization in it Use the CAD parameters Work directly with relevant shapes .

Parametric shape optimization Intrinsic dimensionality Meta-modeling in eigenbasis Optimization in eigenbasis Conclusions Shape space dimension CAD parameters vector x ∈ R d but δ < d dimensions should approximate Ω Ω Ω := { Ω x , x ∈ X } precisely enough. NACA 22 airfoils The x j ’s have an heterogeneous and local influence on Ω x . Consider the designs in a high-dimensional feature space Φ via φ : X − → Φ.

Parametric shape optimization Intrinsic dimensionality Meta-modeling in eigenbasis Optimization in eigenbasis Conclusions Shape space dimension CAD parameters vector x ∈ R d but δ < d dimensions should approximate Ω Ω Ω := { Ω x , x ∈ X } precisely enough. NACA 22 airfoils The x j ’s have an heterogeneous and local influence on Ω x . Consider the designs in a high-dimensional feature space Φ via φ : X − → Φ. Good choice of φ ⇒ possible to retrieve a δ -dimensional manifold.

Parametric shape optimization Intrinsic dimensionality Meta-modeling in eigenbasis Optimization in eigenbasis Conclusions High-dimensional shape mapping Φ ⊂ R D : high-dimensional space of shape representation.

Parametric shape optimization Intrinsic dimensionality Meta-modeling in eigenbasis Optimization in eigenbasis Conclusions High-dimensional shape mapping Φ ⊂ R D : high-dimensional space of shape representation. Three φ ’s and their ability to distinguish the δ -dimensional manifold in the corresponding Φ compared: Characteristic function χ (Ω x ) [Raghavan et al., 2013], Signed distance to contour ε ( ∂ Ω x ) [Raghavan et al., 2014], Discretization of contour D ( ∂ Ω x ) [Stegmann and Gomez, 2002].

Parametric shape optimization Intrinsic dimensionality Meta-modeling in eigenbasis Optimization in eigenbasis Conclusions High-dimensional shape mapping Φ ⊂ R D : high-dimensional space of shape representation. Three φ ’s and their ability to distinguish the δ -dimensional manifold in the corresponding Φ compared: Characteristic function χ (Ω x ) [Raghavan et al., 2013], Signed distance to contour ε ( ∂ Ω x ) [Raghavan et al., 2014], Discretization of contour D ( ∂ Ω x ) [Stegmann and Gomez, 2002].

Parametric shape optimization Intrinsic dimensionality Meta-modeling in eigenbasis Optimization in eigenbasis Conclusions PCA on high-dimensional shape representation Sample N x ( i ) ∼ U ( X ) and build Φ Φ ( i ) = φ ( x ( i ) ) in rows. Φ ∈ R N × D , Φ Φ Φ Φ: the “eigenshapes” v j form Apply a PCA on Φ Φ an orthonormal shape basis { v 1 , . . . , v D } of Φ with decreasing importance. Φ ( i ) = φ φ + � D j =1 α ( i ) j v j ⇒ consider the eigenbasis coordinates φ ( x ( i ) ) = Φ Φ φ Φ ( i ) − φ α ( i ) := ( α ( i ) 1 , . . . , α ( i ) D ) ⊤ = V ⊤ (Φ α α Φ φ φ ). The first α j ’s describe the shapes globally as opposed to the x j ’s ⇒ an opportunity to distinguish a lower-dimensional manifold in Φ?

Parametric shape optimization Intrinsic dimensionality Meta-modeling in eigenbasis Optimization in eigenbasis Conclusions PCA on high-dimensional shape representation Sample N x ( i ) ∼ U ( X ) and build Φ Φ ( i ) = φ ( x ( i ) ) in rows. Φ ∈ R N × D , Φ Φ Φ Φ: the “eigenshapes” v j form Apply a PCA on Φ Φ an orthonormal shape basis { v 1 , . . . , v D } of Φ with decreasing importance. Φ ( i ) = φ j v j ⇒ consider the eigenbasis coordinates φ + � D j =1 α ( i ) φ ( x ( i ) ) = Φ Φ φ α ( i ) := ( α ( i ) D ) ⊤ = V ⊤ (Φ Φ ( i ) − φ 1 , . . . , α ( i ) α Φ φ α φ ). The first α j ’s describe the shapes globally as opposed to the x j ’s ⇒ an opportunity to distinguish a lower-dimensional manifold in Φ?

Parametric shape optimization Intrinsic dimensionality Meta-modeling in eigenbasis Optimization in eigenbasis Conclusions PCA on high-dimensional shape representation Sample N x ( i ) ∼ U ( X ) and build Φ Φ ( i ) = φ ( x ( i ) ) in rows. Φ ∈ R N × D , Φ Φ Φ Φ: the “eigenshapes” v j form Φ Apply a PCA on Φ an orthonormal shape basis { v 1 , . . . , v D } of Φ with decreasing importance. j v j ⇒ consider the eigenbasis coordinates Φ ( i ) = φ φ + � D j =1 α ( i ) φ ( x ( i ) ) = Φ Φ φ α ( i ) := ( α ( i ) D ) ⊤ = V ⊤ (Φ Φ ( i ) − φ 1 , . . . , α ( i ) α α Φ φ φ ). The first α j ’s describe the shapes globally as opposed to the x j ’s ⇒ an opportunity to distinguish a lower-dimensional manifold in Φ?

Parametric shape optimization Intrinsic dimensionality Meta-modeling in eigenbasis Optimization in eigenbasis Conclusions PCA on high-dimensional shape representation Sample N x ( i ) ∼ U ( X ) and build Φ Φ ( i ) = φ ( x ( i ) ) in rows. Φ ∈ R N × D , Φ Φ Φ Φ: the “eigenshapes” v j form Φ Apply a PCA on Φ an orthonormal shape basis { v 1 , . . . , v D } of Φ with decreasing importance. j v j ⇒ consider the eigenbasis coordinates Φ ( i ) = φ φ + � D j =1 α ( i ) φ ( x ( i ) ) = Φ Φ φ α ( i ) := ( α ( i ) D ) ⊤ = V ⊤ (Φ Φ ( i ) − φ 1 , . . . , α ( i ) α α Φ φ φ ). The first α j ’s describe the shapes globally as opposed to the x j ’s ⇒ an opportunity to distinguish a lower-dimensional manifold in Φ?

Parametric shape optimization Intrinsic dimensionality Meta-modeling in eigenbasis Optimization in eigenbasis Conclusions PCA on high-dimensional shape representation Sample N x ( i ) ∼ U ( X ) and build Φ Φ ( i ) = φ ( x ( i ) ) in rows. Φ ∈ R N × D , Φ Φ Φ Φ: the “eigenshapes” v j form Φ Apply a PCA on Φ an orthonormal shape basis { v 1 , . . . , v D } of Φ with decreasing importance. j v j ⇒ consider the eigenbasis coordinates Φ ( i ) = φ φ + � D j =1 α ( i ) φ ( x ( i ) ) = Φ Φ φ α ( i ) := ( α ( i ) D ) ⊤ = V ⊤ (Φ Φ ( i ) − φ 1 , . . . , α ( i ) α α Φ φ φ ). The first α j ’s describe the shapes globally as opposed to the x j ’s ⇒ an opportunity to distinguish a lower-dimensional manifold in Φ? Remark: “Reversed” Kernel PCA. PCA in ad-hoc feature space Φ accessed through input related φ (unknown k ).