=Paper=
{{Paper
|id=Vol-3770/paper7
|storemode=property
|title=Active Learning with Physics-Informed Graph Neural Networks on Unstructured Meshes
|pdfUrl=https://ceur-ws.org/Vol-3770/paper7.pdf
|volume=Vol-3770
|authors=Jens Decke,Alexander Heinen,Bernhard Sick,Christian Gruhl
|dblpUrl=https://dblp.org/rec/conf/ial/DeckeHSG24
}}
==Active Learning with Physics-Informed Graph Neural Networks on Unstructured Meshes==
https://ceur-ws.org/Vol-3770/paper7.pdf
Active Learning with Physics-Informed Graph Neural
Networks on Unstructured Meshes
Jens Decke1,β , Alexander Heinen1 , Bernhard Sick1 and Christian Gruhl1
1
Intelligent Embedded Systems, University of Kassel, 34121 Kassel, Germany
Abstract
This paper investigates the use of Physics-Informed Neural Networks (PINNs) in active learning cycles. We defined
two scenarios: one initially unsupervised and the other initially supervised. PINNs emphasize the integration
of physical laws into neural networks to improve the predictive performance of vanilla neural networks and to
enhance the efficiency of traditional methods for solving partial differential equations (PDEs). Key contributions
include adapting existing computational frameworks to enable the use of Graph Neural Networks for solving
problems that require the calculation of gradients on unstructured triangle meshes, a query strategy focusing
on the physical loss, and a comparative analysis of this strategy against random sampling across both defined
scenarios. This work establishes a foundation for future research aimed at expanding the application of Physics-
Informed Graph Neural Networks (PIGNN) using active learning and addressing real-world problems in fluid
dynamics and electrodynamics.
Keywords
Physics Informed Neural Network, Graph Neural Network, Active Learning
1. Introduction
Solving partial differential equations (PDEs) is of paramount interest in numerous fields of science
and engineering, as they form the foundation for modeling a wide range of physical phenomena.
PDEs describe the behavior of physical systems over space and time, governing processes such as
heat transfer [1], fluid dynamics [2], structural mechanics [3], and electromagnetics [4]. Accurate and
efficient solutions to PDEs are crucial for advancing research and development in these areas, making
them a focal point of computational and analytical studies [5]. Traditional methods for solving PDEs,
such as finite element (FEM), finite difference, and finite volume methods, can be computationally
intensive, especially for high-dimensional problems and complex geometries. In recent years, Physics-
Informed Neural Networks (PINNs) have emerged as a powerful alternative computational framework
that integrates machine learning with fundamental physical laws to address these challenges [6, 7].
By embedding physical constraints directly into the neural networkβs loss function πΏπ‘ππ‘ππ cf. Eq. (1),
they utilize both data loss πΏπππ‘π , cf. Eq. (2) and physics loss πΏπππ , cf. Eq. (3) components. With π as the
weighting factor for the data component of the total loss and N vertices.
πΏπ‘ππ‘ππ = π β
πΏπππ‘π + πΏπππ , (1)
π
1
πΏπππ‘π = β(π’ β π’π,π‘ππ’π )2 , (2)
π π=1 π
πΏπππ = π
(ππ·πΈ) (3)
PINNs offer several advantages over traditional methods and ensure that the solutions are not only
data-consistent but also physically accurate. Additionally, PINNs can naturally incorporate multi-
physics problems and seamlessly handle high-dimensional spaces, providing a flexible and efficient
IAL@ECML-PKDDβ24: 8th Intl. Worksh. & Tutorial on Interactive Adaptive Learning, Sep. 9th , 2024, Vilnius, Lithuania
β
Corresponding author.
Envelope-Open jdecke@uni-kassel.de (J. Decke); alexander.heinen@uni-kassel.de (A. Heinen); bsick@uni-kassel.de (B. Sick);
cgruhl@uni-kassel.de (C. Gruhl)
GLOBE https://www.uni-kassel.de/eecs/ies/ (B. Sick)
Orcid 0000-0002-7893-1564 (J. Decke); 0000-0001-9467-656X (B. Sick); 0000-0001-9838-3676 (C. Gruhl)
Β© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
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Jens Decke et al. CEUR Workshop Proceedings 68β76
approach to solving complex PDEs. In Fig. 1 an active learning (AL) cycle with a PINN as Model is
depicted. The queries from the Selector are directed towards an Oracle which is in our case a FEM
simulation. The AL cycle uses Eq. (1) where πΏπππ‘π measures the mean squared error between a predicted
π’ and a true π’π‘ππ’π solution variable (for instance, the prediction of the electric potential), and is therefore
trained in a supervised manner. While the physics loss Eq. (3) πΏπππ corresponds to the residual π
(ππ·πΈ)
and ensures adherence to the PDE. This loss term operates solely unsupervised on the predicted solution
variable π’. This integration allows PINNs to handle sparse data effectively, making them particularly
useful in real-world applications where data is limited [6].
We use this AL cycle to train the PINN starting from two different initial states. In Scenario U the
model is initially trained completely unsupervised, using the physics-informed loss (3) only, and then
data is provided by the oracle to continue the following iterations using πΏπ‘ππ‘ππ on the additional data
to support the unsupervised training. In Scenario S, we use ground-truth data for supervised training
right from the start and therefore use πΏπ‘ππ‘ππ as a loss function.
The choice of mesh plays a crucial role in the implementation of PINNs, as it defines the discretization
of the problem domain. Structured meshes, with their equidistantly distributed cells, offer computational
efficiency and simplicity by enabling straightforward application of automatic differentiation algorithm
to compute spatial gradients [8]. In contrast, unstructured meshes provide the flexibility to handle
complex geometries and allow for adaptivity in regions requiring higher resolution. Graph Neural
Networks (GNNs) are ideal for solving PDEs on unstructured meshes because they adeptly handle
the complex, irregular topologies of these meshes by learning node relationships directly. However,
the computation of spatial gradients in unstructured meshes is more complex due to the irregular
neighborhoods of their triangular cells. The design of the mesh significantly affects the distribution
Figure 1: An active learning loop for training Physics-Informed Neural Networks (PINNs) utilizing a physics loss-
based query strategy, with FEM simulations as oracle. Highlighting the differences between the two scenarios:
Scenario U operates without ground-truth values and hence unsupervised in the initial training of the model,
contrasting to Scenario S, which follows a traditional supervised active learning approach.
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Jens Decke et al. CEUR Workshop Proceedings 68β76
of data points, the precision of differential operator evaluations, and the enforcement of boundary
conditions. For this reason, we combine a GNN with a physics-informed loss function to develop a
Physics-Informed Graph Neural Network PIGNN. For that, we have to adapt an existing TensorFlow
library to enable the computation of field gradients on unstructured meshes in our PyTorch model.
In summary, PINNs represent a sophisticated method for solving PDEs by integrating neural networks
with physical laws. Enhancing these networks through AL by physical loss residuals, rather than explicit
uncertainty quantification, allows for a more straightforward yet effective refinement process. Strategic
mesh design further augments the model, making PIGNNs a versatile tool for a wide range of applications
in science and engineering.
Contributions
1. We adapt an existing TensorFlow implementation to calculate field gradients on two-
dimensional triangle meshes for our PyTorch model. We use this implementation to build a
physical loss function representing the Poisson equation with Dirichlet boundary conditions
on an unstructured mesh.
2. We propose a simple yet effective query strategy utilizing the physical loss function.
3. We develop and evaluate two distinct PINN-based active learning scenarios, initially unsu-
pervised and initially supervised, comparing our query strategy with random sampling.
The remainder of this article is structured as follows: in Section 2 we summarize the related work
before we introduce our methodology in Section 3. Our preliminary results are presented in Section 4.
The article concludes with a summary of our findings and an outlook for future work in Section 5.
2. Related Work
In this section, we propose related work to the topics of GNNs and PINNs as well as AL and PINNs.
GNNs and PINNs: Solving mesh-based PDEs with neural networks is an increasingly progressive
topic of research. Typical data-driven solution methods come from the fields of computer vision and
graph-based learning [2]. However, these methods lack information about the underlying physics of
the problems at hand.
Initial studies have demonstrated that combining GNNs and PINNs yields excellent results in various
scientific and engineering applications. GNNs excel at processing data represented as graphs, which
is particularly useful for handling complex relationships in unstructured meshes [9, 10]. To leverage
PINNs on unstructured meshes there is an existing package [11] which was initially developed for
TensorFlow. In that way, the capabilities of GNNs can be effectively utilized to design PINNs with the
ability to solve equations containing field gradients.
AL and PINNs: AL for regression tasks is highly effective in reducing the computational load
associated with simulating PDEs. By strategically selecting the most informative samples for extensive
simulation, AL can significantly enhance efficiency [12, 13]. However, for specific applications like
design optimization, where the goal is to systematically identify the optimal design parameters that
satisfy specified performance criteria, it is essential to customize the query strategies. This customization
ensures that iterative algorithms effectively find the best design with minimal PDE evaluations, aligning
the AL process with the optimization objectives and constraints of the physical system described by
PDEs [14].
The idea of combining PINNs with AL is gaining increasing attention. Recent works have taken
initial steps in this direction, employing uncertainty sampling via Monte Carlo dropout [15] to select
informative samples. Another study proposed an adaptive sampling strategy based on Christoffel
functions [16]. In contrast to these approaches, our work focuses exclusively on a score-sampling
strategy based on the physical loss.
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3. Methodology
Our methodology is structured as follows: first, we introduce the data derived from the Poisson equation,
which is a second-order PDE. Subsequently, we present our model, query strategy, and oracle. Finally,
we present our experimental setup.
Data: As dataset we use the charge density input array and the FEM simulated solutions of the Poisson
equation together with the mesh, featuring a circular bounded domain (Ξ© β βπ ), and the associated edge
indices. As input scalar field π we use a random distribution of circular areas with randomly chosen
radii. Although the Poisson equation can be applied to a variety of physics problems, our goal is to
calculate the electric potential field π’ of a given constant charge density distribution π, represented by
the circle areas which is expressed in Eq. (4). Here Ξ represents the Laplacian operator:
βΞπ’ = π in Ξ©
(4)
π’=0 on πΞ©
In this equation, πΞ© denotes the boundary of the domain Ξ©. In Fig. 2, the input features (Fig. 2a) and
the ground-truth solution (Fig. 2b) of a random sample are exemplarily illustrated.
As illustrated in Fig. 2c, we employ an unstructured triangular mesh to discretize the domain. This
type of mesh allows us to accurately capture the geometry and boundary conditions of complex domains.
The physical loss πΏππ·πΈ of the Poisson equation (Eq. (4)) is defined in Eq. (5):
βΞπ’ + π β2 π’ ππ Ξ©
πΏππ·πΈ = π
(ππ·πΈ) = { 2 (5)
βπ’β π’ ππ πΞ©
To compute this loss, it is necessary to obtain the second spatial derivative, indicated by the Laplace
operator. This computation requires considering the spatial dependencies of the mesh cells. While the
Automatic Differentiation (AD) algorithm [8, 17] is typically used for uniform and structured meshes,
it cannot be applied to unstructured meshes used in our study because it struggles with efficiently
propagating derivatives through the complex and irregular connections. Therefore, specialized tech-
niques are needed to handle the unstructured nature of the mesh and accurately compute the required
gradients for the physical loss.
Model: We utilize our PIGNN to efficiently handle the intricate geometries of the domain. The
GNNβs structure is particularly well-suited for capturing the relationships and dependencies within
unstructured data. As GNN type we chose six chebyshev spectral graph convolutional (ChebConv)
layers as the main model and two feed-forward layers as encoder and decoder. The ChebConv layers
(a) Features (b) Ground-Truth (c) Triangular mesh
Figure 2: Images of the input features in (a), the ground-truth solution provided by a FEM simulation in (b)
and the triangular mesh on the circular domain in (c)
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π-hop convolutional operator aggregates information of vertices that are in a radius of π-hops from
the central vertex in contrast to the more popular 1-hop graph convolutional layers, which only take
into account directly connected nodes. Using a π-factor of six allows our model to recognize bigger
structures and helps to minimize the prediction error. To enhance the modelβs capability in dealing with
complex mesh geometries, we integrate it with the MeshGradientPy package [11], which computes
field gradient estimates on every cell based on linear interpolation and then uses an averaging method
to obtain gradient values on vertices. This integration is crucial for accurately resolving the Laplacian,
as specified in Eq. (5). By doing so, we can effectively calculate the unsupervised physical loss πΏππ·πΈ ,
ensuring that the model adheres to the underlying physical laws governing the problem domain. Since
the package is developed for Tensorflow, we adapted the implementation for integration with PyTorch.
Query Strategy: During inference, we can compute the physics residuals π
(ππ·πΈ) without needing
ground-truth values. These residuals are derived from the physical loss πΏππ·πΈ , highlighting samples of
the PIGNNβs predictions that deviate from expected physical behavior. To improve the performance of
our PIGNNs, we employ an innovative strategy that leverages the physical loss πΏππ·πΈ during inference
to guide AL and retraining.
In Eq. (6) our query strategy is depicted. Let π be the set of all samples π₯ that are inferred, and π be
a subset of π containing the π samples with the highest πΏπππ values. The subset π is forwarded to the
Oracle for target value acquisition.
π = {π₯ β π β£ πΏπππ (π₯) β Topπ (πΏπππ )} (6)
In contrast to uncertainty sampling in AL, this strategy works by evaluating the physical residuals,
which quantify how well the predicted solution variable π’ adheres to the governing physical laws; thus,
no additional uncertainty estimation method is required. By identifying samples where the modelβs
predictions are less reliable, we can target specific areas for model improvement. The advantage of this
approach is that we can quantify the physical loss in an unsupervised manner, thereby eliminating the
need for costly epistemic uncertainty quantification methods [18].
This unsupervised quantification of physical loss simplifies the AL process, allowing the model to
autonomously identify and focus on regions with high residuals. These high-residual areas indicate
where the modelβs predictions are most inaccurate, guiding the addition of new data points or retraining
efforts to these critical areas. This method not only streamlines the training process but also ensures
robust model enhancement by continuously refining the model based on its internal assessments of
physical law adherence. This approach is particularly valuable in scenarios where obtaining ground-
truth data is expensive or impractical, as it maximizes the use of available information to improve model
performance and reliability.
Oracle: Focusing on samples with high physical residuals, the Oracle generates additional data in
these regions, thereby improving the modelβs performance. The Model uses its internal physics-based
evaluations to guide its learning process, leveraging both the supervised and unsupervised capabilities
of the PIGNN to ensure its predictions remain physically consistent. The Selector identifies high-residual
samples and the Oracle provides the corresponding true values, which are then included in the training
of the Model for fine-tuning. This active interaction between the Oracle and the Model allows for
targeted improvements in areas where the modelβs predictions are less reliable enhancing the modelβs
performance cost-effectively.
Experimental Setup: Our experimental setup is designed to evaluate two distinct scenarios and is
depicted in Fig. 1:
In Scenario U, we start with a pool of 1500 samples with ground-truth solution data determined
from FEM simulations (oracle). The PIGNN is initially trained on 600 randomly selected samples in
an unsupervised manner using the physics-based loss function πΏπππ (cf. Eq. (3)) only, therefore, no
ground-truth data is provided. After this initial training phase, the model is evaluated on the remaining
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Jens Decke et al. CEUR Workshop Proceedings 68β76
Figure 3: Comparison of our defined Scenarios. Scenario U (orange) is an AL experiment starting with a model
that was initially trained unsupervised whereas in Scenario S the initial state of the model was achieved by
supervised training.
900 samples, calculating physics residuals to identify the 60 samples with the highest residuals (cf.
Eq.(6)). The ground-truth values for these high-residual samples are then obtained from the Oracle
and added to the training set, enabling the use of the total loss Eq. (1) on these additionally acquired
samples. This iterative process of identifying and adding 60 high-residual samples continues until the
cycle iterates 5 times. This scenario is termed unsupervised since the majority of the training is based
on the unsupervised physical loss (cf. Eq. (3)) only, except for the samples added by the oracle.
In Scenario S, we defined a supervised scenario, therefore, using the ground-truth data of the initial
600 randomly selected samples. The total loss πΏπ‘ππ‘ππ (cf. Eq. (1)) is applied to both, the initial samples
and the samples acquired over five iterations, which are provided by the oracle.
For both scenarios, after each iteration, the PIGNN is tested on a separate test dataset of 1500 samples
to evaluate its prediction performance and adherence to physical laws. Additionally, we compare these
methods to a random selection strategy, where 60 random samples are acquired for the training set in
each iteration. This comparison assesses the efficiency of the proposed selection strategy guided by the
physical loss πΏπππ .
4. Preliminary Results and Discussion
The results of our experiments, are summarized in Fig. 3 and will be discussed in the following: First,
we can observe that our proposed query strategy outperforms the random strategy in both scenarios. It
is evident that after the first iteration, Scenario S, which was trained supervised to optimize the total
loss πΏπ‘ππ‘ππ , surpasses Scenario U, where the model was trained solely using the unsupervised loss πΏπππ .
However, after four AL cycles, Scenario U demonstrates superior performance compared to Scenario S.
This indicates that the initial unsupervised training is a viable approach for our PIGNN. Considering
that substantial resources are saved by determining the ground-truth values for the initial training pool
β which in our example involved 600 samples β and given that one FEM simulation in industrial use
cases can take days or even weeks of computing time, the advantages become even more apparent. An
adaptive approach that only simulates the most valuable samples presents significant benefits.
However, an in-depth analysis of the consistency of multiple runs using various seeds was beyond
the scope of this work. Additionally, we did not conduct any hyper-parameter tuning or investigate
AL parameters such as the initial pool size, the acquisition size, or the total budget. Consequently,
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Jens Decke et al. CEUR Workshop Proceedings 68β76
the observed fluctuations in the results may be attributed to these factors. These fluctuations may be
primarily due to the limited number of experiments performed and the non-optimized hyperparameters,
which were not adjusted due to the significant effort required, especially in the context of active
learning [19].
Other typical AL query strategies were also not considered. Another critical parameter is π, which
serves as the weighting factor between the two components of the loss function (cf. Eq. (1)). An
incorrectly chosen π can lead to the optimization being dominated by one part of the loss function,
either πΏπππ‘π or πΏπππ , at the expense of the other. These aspects need to be elaborated in future work.
Electric potential u [V]
0.000 0.002 0.004 0.006 0.008 0.01
β2
β
10
(a) Ground-Truth (b) Prediction (c) L1-error
Figure 4: Random sample from Scenario U. (a) the ground-truth data, (b) the PIGNNβs prediction, (c) the
absolute difference, e.g. L1-error between prediction and ground-truth.
In Fig. 4 a randomly chosen test sample of the final iteration of Scenario U is depicted. It shows that
our AL strategy in combination with the PIGNN is capable of providing high-performing predictions in
Fig. 4b. In Fig. 4c the absolute deviation e.g. the L1-error between the prediction and the ground-truth
solution is depicted.
5. Conclusion, Limitations and Future Work
Our experiments show that our PIGNN is generally suitable for use in AL scenarios. Our proposed
query strategy is built upon the networkβs physical loss, which can be evaluated unsupervised. In
future work, we aim to apply our methodology and model to real-world problems and more complex
datasets from the field of fluid dynamics [20] and electrodynamics [4]. Further, we plan to investigate
other acquisition sizes, total budgets, and the initial selection of samples as well as optimization of
hyperparameters which is in general not trivial in deep AL.
Currently, our PIGNN is validated on a circular problem domain solving the Poisson equation on an
unstructured triangular mesh. In the future, we plan to employ this model for more complex geometries
and physical problems. For the above-mentioned datasets, we intend to solve the Maxwell equations on
an unstructured mesh for modeling an electric motor and address turbulent flow in a U-bend applying
the Navier-Stokes equations on a graded mesh. Another work compares methods from the fields of
computer vision and graph learning on these two datasets [2]. We aim to extend this comparison to
include PINNs. These advancements will help validate the robustness and versatility of our PIGNN
in solving a wider range of complex real-world problems. Furthermore, we want to contribute with
the help of AL to face the problems of data scarcity in the realm of solving computationally expensive
PDEs.
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Jens Decke et al. CEUR Workshop Proceedings 68β76
Acknowledgment
This research has been funded by the Federal Ministry for Economic Affairs and Climate Action (BMWK)
within the project βKI-basierte Topologieoptimierung elektrischer Maschinen (KITE)β (19I21034C).
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