GenPANIS: A Latent-Variable Generative Framework for Forward and Inverse PDE Problems in Multiphase Media

Matthaios Chatzopoulos; Phaedon-Stelios Koutsourelakis

stat.MLFeb 16, 2026

GenPANIS: A Latent-Variable Generative Framework for Forward and Inverse PDE Problems in Multiphase Media

Authors: Matthaios Chatzopoulos, Phaedon-Stelios Koutsourelakis

Organizations: Technical University of Munich, Professorship of Data-driven Materials Modeling, School of Engineering and Design, Boltzmannstr. 15, 85748 Garching, Germany · Munich Data Science Institute (MDSI - Core member), Garching, Germany

Abstract

Inverse problems and inverse design in multiphase media, i.e., recovering or engineering microstructures to achieve target macroscopic responses, require operating on discrete-valued material fields, rendering the problem non-differentiable and incompatible with gradient-based methods. Existing approaches either relax to continuous approximations, compromising physical fidelity, or employ separate heavyweight models for forward and inverse tasks. We propose GenPANIS, a unified generative framework that preserves exact discrete microstructures while enabling gradient-based inference through continuous latent embeddings. The model learns a joint distribution over microstructures and PDE solutions, supporting bidirectional inference (forward prediction and inverse recovery) within a single architecture. The generative formulation enables training with unlabeled data, physics residuals, and minimal labeled pairs. A physics-aware decoder incorporating a differentiable coarse-grained PDE solver preserves governing equation structure, enabling extrapolation to varying boundary conditions and microstructural statistics. A learnable normalizing flow prior captures complex posterior structure for inverse problems. Demonstrated on Darcy flow and Helmholtz equations, GenPANIS maintains accuracy on challenging extrapolative scenarios - including unseen boundary conditions, volume fractions, and microstructural morphologies, with sparse, noisy observations. It outperforms state-of-the-art methods while using 10 - 100 times fewer parameters and providing principled uncertainty quantification.

Figures & tables

Symbol	Definition / Meaning
$\mathbf{s}$	Spatial location in problem domain $\Omega\subset\mathbb{R}^{n}$
$u(\mathbf{s})$	Physical state/PDE solution (e.g., temperature, pressure)
$c(\mathbf{s})$	Spatially varying material coefficient taking discrete values $c^{(0)},\dots,c^{(P-1)}$
$\boldsymbol{x}=\{x_{j}\}_{j=1}^{d_{x}}$	Discrete microstructure vector with $x_{j}\in\{0,1,\dots,P-1\}$ (e.g., $x_{j}\in\{0,1\}$ for two-phase)
$\mathbf{y}\in\mathbb{R}^{d_{\mathbf{y}}}$	PDE-solution coefficient vector (basis expansion coefficients or NN weights)
$\hat{\mathbf{u}}$	Observed (noisy) PDE measurements, $\hat{\mathbf{u}}=\mathbf{H}(u)+\textrm{noise}$

Table 1: Key symbols used in the GenPANIS framework. Symbols with a hat

\hat{\cdot}

indicate observed or given quantities.

Figure 1: Probabilistic graphical model illustration for the GenPANIS framework. Plates indicate different data types (labeled, unlabeled, virtual). See text for details on model structure and parameterization.

Model	PINO	FunDPS	GenPANIS
Training time (in $hrs$ )	$38.2$	$44.5$	$\mathbf{0.6}$
Total Parameters (in millions)	$13.1$	$183.3$	$\mathbf{2.9}$

Table 2: Training times and total number of trainable parameters for each model when training using

10000

labeled data (Darcy Flow Equation). In contrast to PINO and GenPANIS, FunDPS can utilize only labeled data during training. This is why we train all models exclusively with labeled data for results where comparisons are feasible.

$\dim(\boldsymbol{z})$	$L_{2}$ Relative Error (Forward)	Pixel Accuracy (Inverse)	Total Parameters
20	$8.37\cdot 10^{-2}$	$0.936$	$\mathbf{1.3\cdot 10^{6}}$
30	$5.91\cdot 10^{-2}$	$0.981$	$1.7\cdot 10^{6}$
40	$7.17\cdot 10^{-2}$	$\mathbf{0.985}$	$2.1\cdot 10^{6}$
50	$4.68\cdot 10^{-2}$	$0.976$	$2.5\cdot 10^{6}$
60	$4.33\cdot 10^{-2}$	$\mathbf{0.985}$	$2.9\cdot 10^{6}$
70	$7.7\cdot 10^{-2}$	$\mathbf{0.985}$	$3.3\cdot 10^{6}$

Table 3: Relative error (forward problem), pixel accuracy (inverse problem), and total number of trainable parameters for different values of

\dim(\boldsymbol{z})

in GenPANIS.

Type of prior	$L_{2}$ Relative Error (Forward)	Pixel Accuracy (Inverse)
NVP flow model with 12 layers	$\mathbf{7.80\cdot 10^{-2}}$	$\mathbf{0.945}$
$\mathcal{N}\left(\boldsymbol{0},\boldsymbol{I}\right)$	$2.26\cdot 10^{-1}$	$0.920$

Table 4: Results for the forward and inverse problems obtained when the flow model of section 2.1.1 is used as a prior for the latent variables

\boldsymbol{z}

vs. a standard normal

\mathcal{N}\left(\boldsymbol{0},\boldsymbol{I}\right)

Panel of Figure 2 — Figure 2: Forward problem predictions of the three models.

Figure 3: Reference posterior statistics for Darcy flow obtained from one million HMC samples. Full observations were used with $SNR=100$ .

Panel of Figure 4 — Figure 4: Solving the Darcy flow inverse problem for various levels of noise by observing the full solution field. No additional residual information is used after training the models.

Panel of Figure 5 — Figure 5: Solving the Helmholtz equation inverse problem for various levels of noise by observing the full solution field. No additional residual information is used after training the models.

Panel of Figure 6 — Figure 6: Solving the Darcy flow inverse problem for various levels of noise by observing the full solution field. Additional residuals were used for the PINO and FunDPS models only.

Panel of Figure 7 — Figure 7: Solving the inverse problem for various levels of noise by observing the full solution field. Additional residuals were used for the PINO and FunDPS models only.

Panel of Figure 8 — Figure 8: Solving the Darcy flow inverse problem for various observation grids and $SNR=100$ . No additional residual information is used after training the models.

Panel of Figure 9 — Figure 9: Solving the Helmholtz equation inverse problem for various observation grids and $SNR=100$ . No additional residual information is used after training the models.

Panel of Figure 10 — Figure 10: Solving the Darcy flow inverse problem for various randomly selected observation points and $SNR=100$ . No additional residual information is used after training the models.

Panel of Figure 11 — Figure 11: Solving the Helmholtz equation inverse problem for various randomly selected observation points and $SNR=100$ . No additional residual information is used after training the models.

Figure 12: Performance of the examined models, in terms of $L_{2}$ error $\epsilon$ (forward problem, same instance as Figure 4 , noiseless) and pixelwise accuracy $PA$ (inverse problem, same instance as Figure 4 , $SNR=100$ ), when different number of labeled data is employed during training.

Figure 13: Solving the Darcy flow inverse problem for an out-of-distribution input-output pair. The solution field is noisy ( $SNR=100$ ) and fully observed. No additional residual information is used after training the models.

Figure 14: Solving the Darcy flow inverse problem for out-of-distribution Boundary conditions described in Equation 35 . The solution field is noisy ( $SNR=100$ ) and fully observed. No additional residual information is used after training the models.

Figure 15: Solving the Helmholtz equation inverse problem for out-of-distribution Boundary conditions described in Equation 35 . The solution field is noisy ( $SNR=100$ ) and fully observed. No additional residual information is used after training the models.

Figure 16: Solving the Darcy flow inverse problem for out-of-distribution volume fraction (VF=10%). The solution field is noisy ( $SNR=100$ ) and fully observed. No additional residual information is used after training the models.

Figure 17: Solving the Helmholtz equation inverse problem for out-of-distribution volume fraction (VF=10%). The solution field is noisy ( $SNR=100$ ) and fully observed. No additional residual information is used after training the models.

Panel of Figure 18 — Figure 18: Effect of using unlabeled data on top of a small labeled dataset ( $N_{l}=100$ ). Full observations were used for both the forward and inverse problems, while for the latter, the observation noise was equal to $SNR=100$ .

Figure 19: Predictive performance of the model, as more unlabeled training data are added to $N_{l}=100$ labeled training data.

$N_{l}$ / $N_{v}$	$N_{v}=0$	$N_{v}=100$	$N_{v}=1000$	$N_{v}=10000$
$N_{l}=0$	-	$1.547\cdot 10^{-1}$	$1.189\cdot 10^{-1}$	$0.963\cdot 10^{-1}$
$N_{l}=100$	$1.029\cdot 10^{-1}$	$0.983\cdot 10^{-1}$	$0.967\cdot 10^{-1}$	$0.925\cdot 10^{-1}$
$N_{l}=1000$	$0.839\cdot 10^{-1}$	$0.684\cdot 10^{-1}$	$0.638\cdot 10^{-1}$	$0.553\cdot 10^{-1}$
$N_{l}=10000$	$0.560\cdot 10^{-1}$	$0.428\cdot 10^{-1}$	$0.394\cdot 10^{-1}$	$0.387\cdot 10^{-1}$

Table 5:

L_{2}

error

\epsilon

for solving the forward problem (same instance as 4 ) for different combinations of labeled (

N_{l}

) and virtual (

N_{v}

) data.

$N_{l}$ \| $N_{v}$	$N_{v}=0$	$N_{v}=100$	$N_{v}=1000$	$N_{v}=10000$
$N_{l}=0$	-	$0.813$	$0.849$	$0.854$
$N_{l}=100$	$0.844$	$0.853$	$0.859$	$0.883$
$N_{l}=1000$	$0.873$	$0.906$	$0.918$	$0.922$
$N_{l}=10000$	$0.890$	$0.955$	$0.966$	$0.966$

Table 6: Pixelwise Accuracy for solving the inverse problem of (same instance as 4 ,

SNR=100

) for different combinations of labeled (

N_{l}

) and virtual (

N_{v}

) data.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Layers	Input $\rightarrow$ Output	Parameters
Linear	60 $\rightarrow$ 128	7808
LeakyReLU	—	—
Linear	128 $\rightarrow$ 128	16512
LeakyReLU	—	—
Linear	128 $\rightarrow$ 60	7740
Total	—	32060

Appendix

Table 7: Layers and parameters of the scale network

s_{k}(\cdot)

Layers	Input $\rightarrow$ Output	Parameters
Linear	60 $\rightarrow$ 128	7808
LeakyReLU	—	—
Linear	128 $\rightarrow$ 128	16512
LeakyReLU	—	—
Linear	128 $\rightarrow$ 60	7740
Total	—	32060

Appendix

Table 8: Layers and parameters of the translation network

t_{k}(\cdot)

Component	Parameters
Scale network $s_{k}(\cdot)$	32060
Translation network $t_{k}(\cdot)$	32060
Learned scale parameter $\boldsymbol{\alpha}_{k}\in\mathbb{R}^{60}$	60
Total per coupling layer	64180
Total flow model parameters	770160

Appendix

Table 9: Total parameters of the 12-coupling-layer RealNVP prior.

Layers	Feature Maps	Height $\times$ Width	Parameters
Fully Connected Layer (60 $\rightarrow$ 16641)	—	16641 $\times$ 1	1015101
Reshape	—	129 $\times$ 129	—
Convolution Layer (k3s2p1)	16	65 $\times$ 65	160
Batch Normalization	—	65 $\times$ 65	32
Average Pooling Layer (k2s2)	—	32 $\times$ 32	—
Convolution Layer (k3s1p1)	48	32 $\times$ 32	6960

Appendix

Table 10: Layers and parameters of the neural network

f_{\boldsymbol{\theta}_{f}}(\boldsymbol{z})

Layers	Feature Maps	Height $\times$ Width	Parameters
Input	—	129 $\times$ 129	—
Convolution Layer (k3s2p1)	8	65 $\times$ 65	80
Batch Normalization	—	65 $\times$ 65	16
Average Pooling Layer (k2s2)	—	32 $\times$ 32	—
Convolution Layer (k3s1p1)	24	32 $\times$ 32	1752
Batch Normalization	—	32 $\times$ 32	48

Appendix

Table 11: Layers and parameters of the neural network

g_{\boldsymbol{\xi}_{g}}(\hat{\boldsymbol{x}}^{(i)})

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Abstract

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