Multi-material Multi-physics Topology Optimization with Physics-informed Gaussian Process Priors

Xiangyu Sun; Shirin Hosseinmardi; Amin Yousefpour; Ramin Bostanabad

cs.LGFeb 19, 2026

Multi-material Multi-physics Topology Optimization with Physics-informed Gaussian Process Priors

Authors: Xiangyu Sun, Shirin Hosseinmardi, Amin Yousefpour, Ramin Bostanabad

Organizations: Department of Mechanical and Aerospace Engineering, University of California, Irvine · Department of Civil and Environmental Engineering, University of California, Irvine

Abstract

Machine learning (ML) has been increasingly used for topology optimization (TO). However, most existing ML-based approaches focus on simplified benchmark problems due to their high computational cost, spectral bias, and difficulty in handling complex physics. These limitations become more pronounced in multi-material, multi-physics problems whose objective or constraint functions are not self-adjoint. To address these challenges, we propose a framework based on physics-informed Gaussian processes (PIGPs). In our approach, the primary, adjoint, and design variables are represented by independent GP priors whose mean functions are parametrized via neural networks whose architectures are particularly beneficial for surrogate modeling of PDE solutions. We estimate all parameters of our model simultaneously by minimizing a loss that is based on the objective function, multi-physics potential energy functionals, and design-constraints. We demonstrate the capability of the proposed framework on benchmark TO problems such as compliance minimization, heat conduction optimization, and compliant mechanism design under single- and multi-material settings. Additionally, we leverage thermo-mechanical TO with single- and multi-material options as a representative multi-physics problem. We also introduce differentiation and integration schemes that dramatically accelerate the training process. Our results demonstrate that the proposed PIGP framework can effectively solve coupled multi-physics and design problems simultaneously -- generating super-resolution topologies with sharp interfaces and physically interpretable material distributions. We validate these results using open-source codes and the commercial software package COMSOL.

Figures & tables

Figure 1: PIGP framework for multi-physics topology optimization: The mean functions for the design variable, as well as for the primary and adjoint fields, are parameterized using the multi-output PGCAN architecture, which consists of three modules: (i) feature-space encoding via a convolutional neural network, (ii) feature interpolation at all collocation points, and (iii) decoding through a shallow MLP. The design variable $\bm{\rho}(\bm{x})$ is a vector of volume fractions for the void and all candidate materials at each domain point $\bm{x}$ . The final-layer multi-output from PGCAN for $\bm{\rho}(\bm{x})$ is passed through a $\mathrm{softmax}$ function to enforce the partition of unity of $\bm{\rho}(\bm{x})$ . Gaussian processes are employed on these mean functions to impose the BCs for the primary and adjoint fields, and to enforce the design constraints.

Parameter	Void	Material 1	Material 2	Material 3
$E_{i}$	$10^{-5}$	$0.4$	$0.6$	$1.0$
$\kappa_{i}$	$10^{-5}$	$0.2$	$0.5$	$1.0$
$\hat{\rho}_{i}$	$0$	$0.5$	$0.7$	$1.0$
$\hat{p}_{i}$	$0$	$1.6$	$1.2$	$1.0$

Table 1: Parameters of artificial materials: The Young’s modulus

E

(

\mathrm{M}\mathrm{Pa}

), thermal conductivity

\kappa

(

\mathrm{W}/(\mathrm{m}\mathrm{K})

), discrete physical density

\hat{\rho}_{i}

, and unit-volume price

\hat{p}_{i}

are reported for four phases.

Panel of Figure 2 — Figure 2: Benchmark examples for CM: Four canonical benchmark problems are considered, including the Messerschmitt–Bölkow–Blohm (MBB) beam, cantilever beam, bridge beam, and L-shaped beam. The configurations of the 2D examples are shown in (a)–(d) and their 3D counterparts are presented in (e)–(h).

Example	$H$ ( $\mathrm{mm}$ )	$L$ ( $\mathrm{mm}$ )	$W$ ( $\mathrm{mm}$ )	$f$ ( $\mathrm{N}$ )	$\psi_{m}$	$M_{0}$	$\psi_{p}$	$P_{0}$
MBB beam 2D	100	200	–	0.1	0.3	2 $\times 10^{4}$	0.4	3.2 $\times 10^{4}$
Cantilever beam 2D	100	200	–	0.1	0.3	2 $\times 10^{4}$	0.4	3.2 $\times 10^{4}$
Bridge beam 2D	100	200	–	0.1	0.3	2 $\times 10^{4}$	0.4	3.2 $\times 10^{4}$
L-shape beam 2D	100	100	–	0.1	0.2	6.4 $\times 10^{3}$	0.3	1.024 $\times 10^{4}$
MBB beam 3D	30	60	20	0.1	0.1	3.6 $\times 10^{4}$	0.2	5.76 $\times 10^{4}$
Cantilever beam 3D	30	60	20	0.1	0.1	3.6 $\times 10^{4}$	0.2	5.76 $\times 10^{4}$

Table 2: Dimensions and parameters for CM: The geometric dimensions, applied external forces, prescribed mass or cost fractions, and the corresponding reference mass

M_{0}

and cost

P_{0}

for the four benchmark examples are summarized. All dimensions are given in

\mathrm{mm}

and

f

denotes the magnitude of the applied load (

\mathrm{N}

Example	Ordered SIMP	PolyMat	PIGP 2D		SIMP 3D	PIGP 3D
Example	FE Nodes	Total Nodes	Coarse	Fine	FE Nodes	Coarse	Fine
MBB	(301, 151)	45413	(201, 101)	(401, 201)	(77, 39, 27)	(61, 31, 21)	(92, 47, 32)
Cantilever	(301, 151)	45413	(201, 101)	(401, 201)	(77, 39, 27)	(61, 31, 21)	(92, 47, 32)
Bridge	(151, 151)	22734	(101, 101)	(201, 201)	(76, 76, 20)	(61, 61, 16)	(92, 92, 24)
L-shape	(151, 151)	14701	(101, 101)	(201, 201)	(76, 76, 20)	(61, 61, 16)	(92, 92, 24)

Table 3: Grid resolutions for CM: For the ordered SIMP, the FE node counts

(N_{x},N_{y})

along

L

and

H

dimensions are reported. For the PolyMat, only the total number of FE nodes is provided due to its use of unstructured polygonal meshes. For the SIMP 3D, the FE node counts

(N_{x},N_{y},N_{z})

along the domain dimensions

L

H

, and

W

are reported. For our PIGP framework, the numbers of CPs are listed for both coarse and fine grids:

(N_{x}^{(1)},N_{y}^{(1)})

and

(N_{x}^{(n_{g})},N_{y}^{(n_{g})})

for the 2D cases, and

(N_{x}^{(1)},N_{y}^{(1)},N_{z}^{(1)})

and

(N_{x}^{(n_{g})},N_{y}^{(n_{g})},N_{z}^{(n_{g})})

for the 3D cases.

Figure 3: Designed multi-material topologies for 2D CM: Representative optimized topologies are shown for each method under different constraints. PolyMat accommodates only mass or cost constraint but not both.

Figure 4: Representative designs for 3D CM: For each benchmark, the topologies corresponding to the maximum, median, and minimum compliance values are shown, comparing SIMP 3D in the single-material setting with our PIGP 3D framework under both single- and multi-material settings. The three colors denote the three candidate artificial materials defined in Table 1 . The multi-material designs are subject to both mass and cost constraints.

Panel of Figure 5 — Figure 5: Topology evolution during multi-material CM training: Intermediate topologies at selected epochs are shown along with the history of design objective $L_{C}(\cdot)$ for (a) 2D MBB beam, (b) 2D cantilever beam, (c) 2D bridge beam, and (d) 2D L-shaped beam. Panels (e)–(h) show the corresponding topology evolution for the same four examples in the 3D setting.

Panel of Figure 6 — Figure 6: Effect of different approximation techniques on the robustness of our approach: We compare the designed topologies when the residuals are approximated using either shape functions or FD. FD is less stable and generally produces topologies with higher compliance. Details of the two examples are provided in [ 51 ] .

Panel of Figure 7 — Figure 7: Benchmark examples for 2D and 3D heat conduction optimization: The prescribed temperature at the bottom left corner acts as the heat sink while the entire domain is subjected to uniform volumetric heat generation.

Example	$H$ ( $\mathrm{mm}$ )	$L$ ( $\mathrm{mm}$ )	$W$ ( $\mathrm{mm}$ )	$s$ ( $\mathrm{W}/(\mathrm{mm}^{3})$ )	$\psi_{m}$	$M_{0}$
Heat Conduction 2D	100	200	–	1 $\times 10^{-5}$	0.3	2 $\times 10^{4}$
Heat Conduction 3D	30	60	30	1 $\times 10^{-5}$	0.1	5.4 $\times 10^{4}$

Table 6: Design parameters for heat conduction examples: Geometric dimensions, heat source intensity, prescribed mass fraction, and the reference mass

M_{0}

for heat conduction problems shown in Fig. 7 .

Example	Ordered SIMP 2D	PIGP 2D		SIMP 3D	PIGP 3D
Example	FE Nodes	Coarse	Fine	FE Nodes	Coarse	Fine
Heat Conduction 2D	(301, 151)	(201, 101)	(401, 201)	–	–	–
Heat Conduction 3D	–	–	–	(61, 31, 31)	(41, 21, 21)	(81, 41, 41)

Table 7: Grid resolutions for the heat conduction benchmarks: We report the FE node counts and CP resolutions of the dynamic grids for SIMP and PIGP, respectively.

Figure 8: Representative designs for 2D heat conduction optimization: Both approaches are used for single- and multi-material cases. For each case, we observe the designs corresponding to the maximum, median, and minimum thermal compliance values.

Figure 9: Representative designs for 3D heat conduction optimization: For each setting, the designs corresponding to the maximum, median, and minimum thermal compliance values are shown. SIMP 3D [ 33 ] is used only in the single-material setting, whereas the proposed framework accommodates both single- and multi-material settings.

Panel of Figure 10 — Figure 10: Benchmark examples for compliant mechanism design: Design domains and BCs for (a) the displacement inverter and (b) the compliant gripper. In both examples, the objective is to maximize the output displacement $u_{\mathrm{out}}$ .

Example	Ordered SIMP 2D	PIGP 2D
Example	FE nodes	Coarse	Fine
Displacement Inverter	(201,101)	(201,101)	(401,201)
Compliant Gripper	(201,101)	(201,101)	(401,201)

Table 10: Grid settings for the compliant mechanism benchmarks: We report the FE node counts and CP resolutions of the dynamic grids for SIMP and PIGP, respectively.

Panel of Figure 11 — Figure 11: Representative topologies for compliant mechanism design: Three independent runs corresponding to the maximum, median, and minimum objective values are shown for both the single-material and multi-material settings.

Panel of Figure 12 — Figure 12: Thermo-mechanical compliant device design with the objective of maximizing output displacement: Design domains and BCs for (a) the thermal actuator and (b) the thermal gripper. For both examples, a high-temperature BC is applied along the left edge.

Material	$\alpha\;(\mathrm{K}^{-1})$	$\kappa\;(\mathrm{W}/\upmu\mathrm{m}\mathrm{K})$	$\rho\;(\mathrm{g}/\mathrm{c}\mathrm{m}^{3})$	$E\;(\mathrm{G}\mathrm{Pa})$	$s\;(\mathrm{W}/\upmu\mathrm{m}^{3}\,\mathrm{K})$
Ni	$1.5\times 10^{-5}$	$9.07\times 10^{-5}$	8.9	200	- $4.5\times 10^{-8}$
Fe	$1.2\times 10^{-5}$	$6.00\times 10^{-5}$	7.8	200	- $4.5\times 10^{-8}$
Al	$2.3\times 10^{-5}$	$2.37\times 10^{-4}$	2.7	70	- $4.5\times 10^{-8}$
Cu	$1.7\times 10^{-5}$	$4.00\times 10^{-4}$	8.96	128	- $4.5\times 10^{-8}$
Ti	$8.6\times 10^{-6}$	$2.59\times 10^{-5}$	4.5	120	- $4.5\times 10^{-8}$

Table 13: Material properties for multi-material thermo-mechanical design: Coefficient of thermal expansion

\alpha

, thermal conductivity

\kappa

, physical density

\rho

, Young’s modulus

E

, and heat source

s

for examples in Fig. 12 are presented.

Example	COMSOL	PIGP 2D
Example	FE nodes	Coarse	Fine
Thermal Actuator	(135, 68)	(217, 109)	(277, 139)
Thermal Gripper	(135, 68)	(217, 109)	(277, 139)

Table 14: Grid resolution for the thermo-mechanical design applications: We report the FE node counts and CP resolutions of the dynamic grids for COMSOL and PIGP, respectively.

Panel of Figure 13 — Figure 13: Thermo-mechanical gripper design: A representative solution for the thermal gripper showing (a) the design domain and the optimized topology and (b) the corresponding displacement and temperature fields along with their adjoint counterparts.

Panel of Figure 14 — Figure 14: Thermo-mechanical designs of PIGP vs COMSOL: Optimized topologies for (a) the thermal actuator and (b) the thermal gripper. COMSOL is evaluated under the single-material setting using Ni, whereas PIGP is evaluated under both single- and multi-material settings where in the latter case two material combinations are considered. The visualized topologies correspond to simulations associated with the maximum, median, and minimum objective values.

Panel of Figure 15 — Figure 15: Topology evolution and training dynamics for the thermal gripper: (a) Design objective history with the intermediate topologies at selected epochs. (b) Evolution of the five loss components including the potential energy functionals for the displacement and temperature fields and their adjoint fields, along with the mass-fraction residual. (c) Evolution of the gray area fraction and the scheduled mass fraction. Similar trends are observed for the other thermo-mechanical examples.

Panel of Figure 16 — Figure 16: Energy histories for the displacement, temperature, and adjoint fields: Evolution of (a) the stored strain energy and spring energy for the primal displacement field, (b) the stored strain energy, spring energy, and external work for the adjoint displacement field, (c) the thermal energy and source-loss energy for the temperature field, and (d) the thermal energy and source energy for the adjoint temperature field.

Topology Optimization

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HPG-Diff: Hierarchical physics-guided diffusion with differentiable connectivity constraints for topology optimization

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Topology Optimization Physics-Informed Diffusion Models Diffusion Model Guidance Physics-Informed ML

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Deep generative models offer a promising paradigm for topology optimization, enabling rapid design exploration. However, these approaches lack intrinsic physics guidance, often leading to poor generalizability across unseen boundary conditions and the formation of floating material artifacts. To address these limitations, we propose Hierarchical Physics-Guided Diffusion (HPG-Diff), a novel diffusion framework that enforces physics consistency through two synergistic mechanisms. First, we introduce a hierarchical physics-guided strategy that aligns different precomputed physics features with the denoising process, guiding material distribution toward optimal load paths to enhance generalizability. Second, we propose a floating material suppression loss as a differentiable connectivity constraint inspired by thermal conduction to improve topological connectivity. By simulating a virtual heat propagation process from load positions, this mechanism explicitly penalizes floating material during training. Quantitative evaluations demonstrate that HPG-Diff achieves average compliance errors of 0.87% (in-distribution) and 5.29% (out-of-distribution), while reducing floating material ratios to 2.90% and 2.44%, respectively. Furthermore, case studies on a 3:1 rectangular domain, including cantilever and bridge benchmarks, provide preliminary evidence that lightweight LoRA fine-tuning with a small dataset can support the adaptation of HPG-Diff to rectangular non-square domains.

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