Guided Diffusion by Optimized Loss Functions on Relaxed Parameters for Inverse Material Design

Jens U. Kreber; Christian Weißenfels; Joerg Stueckler

cs.LGFeb 17, 2026

Guided Diffusion by Optimized Loss Functions on Relaxed Parameters for Inverse Material Design

Authors: Jens U. Kreber, Christian Weißenfels, Joerg Stueckler

Organizations: Intelligent Perception in Technical Systems Group, University of Augsburg, Germany · Faculty of Mathematics, Natural Science and Engineering, University of Augsburg, Germany · Centre for Advanced Analytics and Predictive Sciences, University of Augsburg, Germany

Abstract

Inverse design problems are common in engineering and materials science. The forward direction, i.e., computing output quantities from design parameters, typically requires running a numerical simulation, such as a FEM, as an intermediate step, which is an optimization problem by itself. In many scenarios, several design parameters can lead to the same or similar output values. For such cases, multi-modal probabilistic approaches are advantageous to obtain diverse solutions. A major difficulty in inverse design stems from the structure of the design space, since discrete parameters or further constraints disallow the direct use of gradient-based optimization. To tackle this problem, we propose a novel inverse design method based on diffusion models. Our approach relaxes the original design space into a continuous grid representation, where gradients can be computed by implicit differentiation in the forward simulation. A diffusion model is trained on this relaxed parameter space in order to serve as a prior for plausible relaxed designs. Parameters are sampled by guided diffusion using gradients that are propagated from an objective function specified at inference time through the differentiable simulation. A design sample is obtained by backprojection into the original parameter space. We develop our approach for a composite material design problem where the forward process is modeled as a linear FEM problem. We evaluate the performance of our approach in finding designs that match a specified bulk modulus. We demonstrate that our method can propose multiple diverse designs within 1% relative error margin from medium to high target bulk moduli in 2D and 3D settings. We also demonstrate that the material density of generated samples can be minimized simultaneously by using a multi-objective loss function.

Figures & tables

Panel of Figure 1 — Figure 1: Overview of our proposed method. Original design parameters are relaxed into a continuous grid representation on which a diffusion model acts as prior. We guide the sample generation process zero-shot by objective functions using implicit differentiation of the FEM solver. Our approach finds diverse samples with low objective cost.

		frac					cov					ent
$K^{*}$		4.8 ′	86.6	168.5 ′	250.4	332.2 ′	4.8 ′	86.6	168.5 ′	250.4	332.2 ′	4.8 ′	86.6	168.5 ′	250.4	332.2 ′
ours	$\epsilon_{r}<1\%$	0.012	0.030	0.147	0.100	0.055	0.026	0.054	0.230	0.081	0.056	1.000	1.928	4.711	3.774	3.056
ours	$\epsilon_{r}<5\%$	0.030	0.187	0.575	0.498	0.255	0.056	0.202	0.496	0.218	0.139	2.362	4.820	6.322	5.593	4.639
ours, proj.	$\epsilon_{r}<1\%$	0.008	0.058	0.233	0.235	0.038	0.018	0.093	0.284	0.157	0.044	0.667	3.187	5.206	4.801	2.399
ours, proj.	$\epsilon_{r}<5\%$	0.038	0.335	0.867	0.683	0.250	0.063	0.254	0.492	0.224	0.165	2.701	5.261	6.766	5.700	4.645
cond. diff †	$\epsilon_{r}<1\%$	0.007	0.058	0.130	0.120	0.062	0.016	0.121	0.214	0.129	0.081	0.667	3.500	4.551	4.238	3.385

Table 1: Evaluation of our method, projection variant and alternative inverse design approaches on the 2D problem on objective

J_{1}

. Metrics are computed over 200 samples each at different target

K^{*}

and averaged over 3 training seeds for each method.

K^{*}

marked with ′ are part of the targets used for (guidance) hyper parameter optimization. † : trained for specific objective; ‡ : trained for specific target values.

Figure 2: Inverse 2D material designs. Generated samples for selected bulk moduli $K^{*}$ , ordered by relative error quantile. Best on the left, worst on the right. Labels show $K_{s}\,/\,K_{\theta}\,/\,\epsilon_{r}$ . The values $(E,\nu,\rho)$ in the normalized coordinate space are encoded as $(r,g,b)$ values of the image. Our model is able to propose diverse and plausible designs close to the target bulk moduli.

Table 2: Performance for guidance by

J_{1},J_{2}

with ablations. Mean metrics over all targets

K^{*}

for 200 samples each and over 3 model training seeds.

Figure 3: Inverse 3D material designs. Generated samples for selected bulk moduli $K^{*}$ , ordered by relative error quantile. Best on the left, worst on the right. Labels show $K_{s}\,/\,K_{\theta}\,/\,\epsilon_{r}$ . The values $(E,\nu,\rho)$ in the normalized coordinate space are encoded as $(r,g,b)$ values of the image. Our model is able to propose diverse and plausible designs close to the target bulk moduli.

	metric: frac					cov					ent
	$K^{*}$ : 0.9	83.3	165.6	248.0	330.3	0.9	83.3	165.6	248.0	330.3	0.9	83.3	165.6	248.0	330.3
$\epsilon_{r}<1\%$	0.003	0.050	0.118	0.188	0.047	0.008	0.097	0.179	0.093	0.036	0.000	3.199	4.402	4.006	1.862
$\epsilon_{r}<5\%$	0.020	0.240	0.510	0.608	0.305	0.040	0.308	0.357	0.224	0.093	1.947	5.260	6.153	5.516	4.022
$\epsilon<1$	0.483	0.057	0.072	0.077	0.013	0.308	0.109	0.127	0.056	0.018	5.789	3.348	3.717	3.170	1.032
$\epsilon<5$	0.792	0.285	0.337	0.328	0.093	0.387	0.337	0.308	0.139	0.044	6.354	5.472	5.666	4.664	2.624
$\epsilon<10$	0.937	0.518	0.593	0.510	0.188	0.419	0.409	0.383	0.202	0.071	6.546	6.240	6.330	5.311	3.417

Table 3: Evaluation of our method on the 3D problem on objective

J_{1}

. Metrics are computed over 200 samples each at different target

K^{*}

and averaged over 3 model training seeds.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: Visualization of base materials used. Color represents the index in the 168 non-empty chunks.

Panel of Figure 5 — Figure 5: Histograms of bulk modulus $K$ of individual examples in datasets. Guidance targets are chosen uniformly spaced between the 1 and 99 percentile.

Panel of Figure 6 — Figure 6: Bin averages of $\Delta K$ (difference to next higher value) of individual examples in datasets.

Figure 7: Difference

\Delta K

of base material’s

K

to next higher value, averaged over

K

bin (20 bins total). Logarithmic

y

scale, cut at

K=450

Figure 8: Relation between quotient of Elastic modulus of matrix and particle (y-axis) and quotient of bulk modulus K and lower (

b^{-}

) / upper (

b^{+}

) Voigt-Reuss bounds ( Zohdi and Wriggers, 2008 ) . Shown are 500 samples generated similarly to our 3D training dataset. The lower bound fraction is clipped at 2 for better display.

$K^{*}$	4.8	86.6	168.5	250.4	332.2
$V_{m}$	0.044	0.002	0.001	0.009	0.008
$d_{m}$	0.118	0.144	0.113	0.224	0.283
frac $\epsilon_{r}<1\%$ sample	0.055	0.048	0.298	0.692	0.677
frac $\epsilon_{r}<5\%$ sample	0.150	0.572	0.985	0.940	0.830
frac $\epsilon_{r}<1\%$ no closest material	0.027	0.077	0.257	0.497	0.527
frac $\epsilon_{r}<5\%$ no closest material	0.123	0.550	0.983	0.862	0.810

Appendix

Table 4: Further analysis of generated samples. Guidance for

J_{1}

on the 2D problem with our main settings, 200 generated samples, averaged over 3 model training seeds. Results in the last two rows are identical to Table 1 .

ds size	$V_{m}\,\downarrow$	$d_{m}\,\downarrow$	cov $\uparrow$
1k	0.0007	0.1074	0.9742
2.5k	0.0007	0.1137	0.9802
5k	0.0013	0.1162	0.9841
10k	0.0011	0.1164	0.9802

Appendix

Table 5: Comparison of different training dataset sizes for the 2D problem. 3 models with different seeds are trained for each row and 1000 samples generated each, averaged. Generation is unguided with 1000 diffusion steps.

Figure 9: Effect of parameters

\rho_{D}

and maximum gradient magnitude on

J_{1}

. Computed over 200 samples, averaged over 3 target bulk moduli. Black dots show evaluated data points, the red cross indicates the value used in our main experiments.

Figure 10: Effect of parameter

\rho_{D}

on all metrics. Mean metrics over all targets

K^{*}

for 200 samples each and over 3 model training seeds. The values at

\rho_{D}=1

are identical to Table 2(a) .

Figure 11: Inverse 2D material designs. Generated samples for selected bulk moduli $K^{*}$ , ordered by relative error quantile. Best on the left, worst on the right. Labels show $K_{s}\,/\,K_{\theta}\,/\,\epsilon_{r}$ . The values $(E,\nu,\rho)$ in the normalized coordinate space are encoded as $(r,g,b)$ values of the image.

Figure 12: Inverse 3D material designs. Generated samples for selected bulk moduli $K^{*}$ , ordered by relative error quantile. Best on the left, worst on the right. Labels show $K_{s}\,/\,K_{\theta}\,/\,\epsilon_{r}$ . The values $(E,\nu,\rho)$ in the normalized coordinate space are encoded as $(r,g,b)$ values of the image.

setting	batch size	total time	FEM solve & diff. (fraction)
2D 32x32	50	387 s	385 s (99.4%)
2D 64x64	50	1241 s	1236 s (99.5%)

Appendix

Table 6: Runtime analysis of our approach. Results are reported for the generation of one batch of the specified size for

J_{1}

, averaged over 5 target bulk moduli, 200 samples total. Guided diffusion with 100 steps (99 gradient computations per sample). The FEM gradient computation clearly dominates the runtime.

		frac					cov					ent
$K^{*}$		4.8 ′	86.6	168.5 ′	250.4	332.2 ′	4.8 ′	86.6	168.5 ′	250.4	332.2 ′	4.8 ′	86.6	168.5 ′	250.4	332.2 ′
ours	$\epsilon_{r}<1\%$	0.008	0.028	0.013	0.035	0.017	0.015	0.053	0.009	0.038	0.015	1.000	1.765	0.123	0.212	0.304
ours	$\epsilon_{r}<5\%$	0.013	0.058	0.013	0.075	0.092	0.019	0.036	0.009	0.095	0.040	0.664	0.385	0.010	0.328	0.426
ours, proj.	$\epsilon_{r}<1\%$	0.003	0.019	0.034	0.049	0.025	0.006	0.028	0.015	0.042	0.024	0.577	0.375	0.175	0.424	0.987
ours, proj.	$\epsilon_{r}<5\%$	0.006	0.066	0.026	0.077	0.111	0.023	0.019	0.038	0.074	0.068	0.371	0.039	0.040	0.536	0.520
cond. diff	$\epsilon_{r}<1\%$	0.006	0.010	0.030	0.035	0.028	0.014	0.012	0.042	0.034	0.025	0.577	0.255	0.295	0.407	0.742

Appendix

Table 7: Standard deviations of Table 1 (comparison of our method, projection variant and alternative inverse design approaches on the 2D problem on objective

J_{1}

), computed over 3 training seeds for each method.

Table 8: Standard deviations of Table 2 (ablations and guidance with

J_{2}

), computed over 3 model training seeds and all 5 target bulk moduli.

	metric: frac					cov					ent
	$K^{*}$ : 0.9	83.3	165.6	248.0	330.3	0.9	83.3	165.6	248.0	330.3	0.9	83.3	165.6	248.0	330.3
$\epsilon_{r}<1\%$	0.003	0.009	0.008	0.060	0.037	0.007	0.012	0.016	0.009	0.021	0.000	0.263	0.117	0.122	1.639
$\epsilon_{r}<5\%$	0.005	0.044	0.036	0.083	0.160	0.009	0.018	0.024	0.027	0.052	0.338	0.265	0.071	0.079	1.105
$\epsilon<1$	0.045	0.018	0.010	0.029	0.013	0.040	0.027	0.009	0.018	0.016	0.005	0.423	0.245	0.403	0.923
$\epsilon<5$	0.053	0.051	0.042	0.103	0.064	0.042	0.012	0.027	0.019	0.028	0.029	0.248	0.134	0.168	1.006
$\epsilon<10$	0.010	0.043	0.057	0.119	0.125	0.045	0.018	0.027	0.010	0.043	0.096	0.127	0.115	0.052	1.133

Appendix

Table 9: Standard deviations of Table 3 (evaluation of our method on the 3D problem), computed over 3 training seeds.

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