Physics-Informed DeepONets for drift-diffusion on metric graphs: simulation and parameter identification
Authors: Jan Blechschmidt
Tom-Christian Riemer
Max Winkler
Martin STOLL
Jan-Frederik Pietschmann
Venue: ICML 2025 poster (2025)
Keywords: physics informed learning
deep operator nets
metric graphs
drift-diffusion
parameter identification
Primary area: deep_learning->algorithms
TL;DR: novel physics informed deep learning approach for solving nonlinear drift-diffusion equations and related inverse problems on metric graphs

Abstract: We develop a novel physics informed deep learning approach for solving nonlinear drift-diffusion equations on metric graphs. These models represent an important model class with a large number of applications in areas ranging from transport in biological cells to the motion of human crowds. While traditional numerical schemes require a large amount of tailoring, especially in the case of model design or parameter identification problems, physics informed deep operator networks (DeepONets) have emerged as a versatile tool for the solution of partial differential equations with the particular advantage that they easily incorporate parameter identification questions. We here present an approach where we first learn three DeepONet models for representative inflow, inner and outflow edges, resp., and then subsequently couple these models for the solution of the drift-diffusion metric graph problem by relying on an edge-based domain decomposition approach. We illustrate that our framework is applicable for the accurate evaluation of graph-coupled physics models and is well suited for solving optimization or inverse problems on these coupled networks.

Physics-Informed DeepONets for drift-diffusion on metric graphs: simulation and parameter identification
Jan Blechschmidt 12
Tom-Christian Riemer'
Max Winkler'
Martin Stoll
Jan-F. Pietschmann'
Abstract
S
We develop a novel physics informed deep learn- ing approach for solving nonlinear drift-diffusion equations on metric graphs. These models repre- an important model class with large number of applications in areas ranging from transport in biological cells to the motion of human crowds. While traditional numerical schemes require a large amount of tailoring, especially in the case of model design or parameter identificatinn prob- lems, physics informed deep operator networks (DEEPONETS) have emerged as a versatile tool for the solution of partial differential equations with the particular advantage that they easily in- corporate parameter identification questions. We here present an approach where we first learn three DEEPONET models for representative in- flow, inner and outflow edges, resp., and then subsequently couple these models for the solu- tion of the drift-diffusion metric graph problem by relying tion approach. We illustrate that our framework is applicable for the accurate evaluation of graph- coupled physics models and well suited for solving optimization or inverse problems on these coupled networks.
S5
S
S1
2/
S2
64
174
C5
63
U1
1/3
Pe
N
Outtlow
2/
2 origin
inner edge DEEPONET
inflow vertex
edge-based domain decomposi-
outflow vertex
inner vertex measurement sensor
1. Introduction
in many application areas. We focus on the case of met- ric graph where each edge is associated with an interval of (possibly) different length. Therefore, the metric graph can be equipped with a differential operator, acting separately on each edge and with appropriate coupling conditions or boundary conditions at the nodes, called the Hamiltonian, leading to what is known as quantum graphs (Lagnese et al., 2012; Berkolaiko & Kuchment, 2013). Numerical methods for quantum graphs have gained recent interest (Arioli & Benzi, 2018; Gyrya & Zlotnik, 2019; Stoll & Winkler, 2021) both for simulation of PDE models as well as for solving de- sign or inverse problems. As the structure of such graphs is typically rather complex, efficient schemes such as domain decomposition methods (Leugering, 2017) are often needed for computational efficiency.
Dynamic processes on graphs (Newman, 2018; Barabási, 2016) are crucial for understanding complex phenomena
'Department of Mathematics, TU Chemnitz, Chemnitz, Germany *Department of Mathematics and Computer Science, TU Bergakademic Freiberg, Freiberg, Germany 'Department of Mathematics, University of Augsburg, Augshurg, Germany and Centre for Advanced Analytics and Predictive Sciences (CAAPS), University of Augsburg, Universitätsstr. 12a, 86159 Augsburg, Germany.
Correspondence to: Jan Blech- schmidt <jan. www.bblhhhhhhcieeecccc mrnath.tu-chemmitz.e Martin Stoll <martin, stoll @ math. tu chemnitz.de>>
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Physics-Informed DeepONets for drift-diffusion on metric graphs
In this paper we propose machine learning technique, namely, the physics-informed DEEPONET approach (Lu et al., 2021; Wang et al., 2021) for drift-diffusion metric graphs. These methods have been introduced to improve on the performance of the, by now well established, physics- informed neural networks (PINNS) (Raissi et al., 2019), which have found their way into many application areas (Zhu et al., 2019; Jin et al., 2021; Sahli Costabal et al., 2020) including fluid dynamics (Raissi al., 2018; 8; Mao et al., 2020; Lye et al., 2020; Magiera et al., 2020; Wes- sels et al., 2020), continuum mechanies and elastodynamics (Haghighat et al., 2020; Nguyen- Seuyen-Thanh et al., 2020; Rao et al., 2020), inverse problems (Meng& & Karniadakis, 2020; Jagtap et al., 2020), fractional Dnivectionddiffuiinn equations (Pang et al., 2019), stochastic equations (Chen et al., 2021), stochastic differential equa- tions (Yang et al., 2020) and power systems (Misyris al., 2020). XPINNs (eXtended PINNs) are introduced in (Jag- tap& & Karniadakis, 2020) as a generalization of PINNS involving multiple neural networks allowing for paralleliza- tion in space and time via domain decomposition, see also (Heinlein al., 2021) for a review on machine learning ap- proaches in domain decomposition. Due to its broad range of applications, the PINN approach helped to establish the field of scientific machine learning (Thiyagalingam et al., 2022; Rackauckas et al., 2020; Cuomo et al., 2022; Blech- schmidt& & Ernst, 2021). On the other hand, the PINN ap- proach often suffers from reduced accuracy when compared with classical numerical methods for differential equations. Furthermore, has to be retrained everytime when initial conditions, boundary conditions parameters of the PDE change. The DEEPONET architecture was introduced based on the universal approximation theorem for operators and re- lies on two neural networks for learning a representation of the solution operator, namely a branch ner for the input vari- ables, e.g., time l and space x, and a second neural network called trunk ner encoding boundary and initial conditions conditions well as other parameters of the underlying problem, e.g., a variable velocity, viscosity or heat conduc- tivity. Similar to XPINNs DEFPONET has been extended for a domain decomposition application (Yin et al., 2022) where the key component is the coupling condition between the different domains that are constructed during the domain partitioning.
However, depending on the coupling conditions at vertices, different type of models have be learned which distin- guishes our approach from classical domain decomposition methods. Once these models are trained, we are able to obtain solutions on virtually arbitrary graphs via a computa- tionally cheap optimization of loss terms at the nodes which ensure the coupling conditions. This advantage becomes even more significant when considering parameter identifi- cation problems where traditional PDE optimization based approaches would require many solutions of forward and adjoint equations, (De los Reyes, 2015). In our setting, solv- ing the inverse problem merely manifests itself in adding additional loss terms. Therefore, strikingly, the cost of solv- ing the forward and the inverse problem are practically the same.
Our main contributions are as follows:
We propose methodology to solve PDEs on graphs us- ing a novel Lego-like domain decomposition approach where graph edges are represented by DEEPONET models.
Graph-agnostic training of the edge surrogate DEEP- ONET model based on inner, inflow and outflow edges. No additional training is required to couple these models for representing flows on arbitrarily complex graphs.
The novel DEEPONET architecture enables robust model evaluation but also allows the solution of opti- mization or inverse problems at almost no additional cost. This is exemplified on parameter identification problem.
2. Drift-diffusion equations on metric graphs
Let us introduce our notion of metric graph in more detail. A metric graph is a directed graph that consists of a set of vertices V and edges E connecting pair of vertices denoted by (vg e²) where vg of E V. Here vg denotes the vertex at the origin while vf denotes the terminal vertex. In contrast to combinatorial graphs a length E, is assigned to each edge e e E. We identify each edge with a one-dimensional inter- val which allows for the definition of differential operators. The graph domain is then denoted by
In this work we introduce the extension of the DEEPONET framework to graphs, particularly the application to the case of drift-diffusion equation posed on a metric graphs. Drift- diffusion models are used in many application areas ranging from modeling electrical networks, (Hinze et al., 2011), to simulation of traffic flow in cities, (Coclite et al., 2005), and thus serve relevant and sufficiently complex test case,
10, E.j.
3E0
We also introduce a normal vector ne(v) defined as (e = -1 and (et) = 1. To prescribe the behav- ior at the boundary of the graph, we first subdivide the set of vertices V into the interior vertices Vi: and the exterior vertices Va as follows:
On the metric graph, the domain is naturally composed of possibly large number of domains, i.c., the different edges.
2

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Physics-Informed DeepONets for drift-diffusion on metric graphs
the set of interior vertices v E Vi C V contains all vertices that are incident to at least one incoming edge and at least one outgoing edge (i.e. Ve E Vx Jer, e2 e E such that ii = and 2h2 v),
where ue ielllow : (0, R4, uy gutllow (0, + R4 VD. are functions prescribing the rate of influx of mass into the graph as well as the velocity of mass leaving the graph at the boundary vertices. Note that this choice ensures that the bounds 0 Σ Pe S 1 are preserved, while the total mass on the complete graph may change over time, In typi- cal situations, boundary vertices are either of influx- or of outflux type, i.c. u. inllloww (t)u (t) = 0 for all v Vp.
the set of exterior vertices e E VD= == V Vi. contains vertices to which either only incoming or only outgoing edges are incident, i.e., either v² = c or 2 = e holds Ve E E, with &, the edge set incident to vertex e.
The Kirchhoff- Seetooff--eemann conditions are the natural boundary conditions for the differential operator (I), as they ensure that mass enters or leaves the system only via the boundary nodes VD for which either uinfoow or gurfoow is positive.
The differential operator defined on each edge consists of the non-linear drift-diffusion equation given by
H(pe) =- -,pe- Pxpe- f(p)) O, EEE, (1)
Having introduced the complete continuous model, we state the following existence and uniqueness result, whose proof can be found in Appendix A, together with detailed defini- tion of the function spaces involved.
where Pe: x (0.7) + R+ describes the concentration of some quantity on the edge E &, Ve > 0 is an edge- dependent velocity and æ > 0 (typically small) diffusion constant. Furthermore, f R++ + R2 satisfies f(0) = f(1) = 0. This property ensures that solutions satisfy 0 S Pe S a.e. on each edge, see Theorem 2.2. By identifying each edge with an interval [0, Ee). we define the flux as
Theorem 2.2. Let the initial data uimit E LT(T) satisfy O S winit < a.e. on & and let nonnegative functions we unlllow u -ugullow E (0, U E VD and non-negative num- bers Ve, e E E, be given. Then there exists a unique weak solution P P (T)) n H (0, (H))((T))))))))) s.t.
Je(x)= == - (x) + Me I (p(()))
(2)
A typical choice for f used in the following is fpe) =
Pe(1 - Pe).
ge + ( (s0ppe(() Σ (( - Ve I (pe(t))) @uspe) de EEE'
Remark 2.1. Note that the choice ve > 0 results in the fact that the prefered direction of transport is encoded in the direction of the edge (on our directed graph). On the other hand, due to the additional diffusion contributions, the flux Je, and thus the direction of mass transport on each edge, may change sign.
+ Σ VEVD
(-u inflow
+u catflow (t)pt u)) = 0,
for all test functions ip E HT((T) and a.a. 1 E (0, Here L denotes the space of square integrable functions. The space H. denotes the space of functions for which also the weak derivative is bounded in L and with (H its dual space. The Bochner spaces contain time-dependent func- tions where for ult, x) to belong to, e.g. (0, H (T)), the norm
To make (1) a well-posed problem, we need to add initial- conditions well as coupling conditions in the vertices. First we impose on each edge e & the following initial condition
Pe (0, x) = uee jnit (x), for almost all x E (0, e E
(3)
L. |lu(t,-)) dz
with init E LL2(e)) (e).
For vertices v E Vi C V. we apply homogeneous Kirchhoff- Neumann conditions, e., there holds
has to be finite.
Σ Je(w)ne(u) =
(4)
3. Learning surrogate models
nEE.
We apply the operator learning approach to obtain models for the dynamics on edges, given initial and boundary data. Due to the boundary and flux conditions (4)-(6) each graph can be partitioned into three types of edges:
for almost every t E (0, and with & the edge set incident to the vertex v. Additionally, we ask the solution to be continuous over vertices, i.c.
Pe(u)= = (v) for all v E VKc e, e' E Evs
(5)
inflow edges originate in a vertex vo E Vo with uinllow (t) of 0 and terminate in a inner vertex,
again for almost every (). In vertices u E VD
V \. Vx the solution P fulfills flux boundary conditions
inner edges originate and terminate in inner vertices,
Σ (v) ne (v) = - u inflow (1) (1 - Pp)) + uh
guttoo
(1) Pe:
outflow edges originate in an inner vertex and terminate vertex vt E VD with u outtoow (t) + 0.
3Ec.
(6)
3

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Physics-Informed DeepONets for drift-diffusion on metric graphs
In our framework we design one DEEPONET model for each of these three different edges. Once trained, this will allow to construct composite model using the DEEP- ONET submodels for inflow, outflow and inner edges as building blocks of typical graphs. To be more pre- cise, the PDE-describing sensor measurements u (ue (urrigin target init Me) are edge-specific, since they have to accommodate for different types of flux conditions, either Kirchhoff-Neumann conditions (4) for inner edges or inflow and outflow conditions (6) for inflow and outflow edges, respectively.
Training graph
Training graph II
Ti Training graph III
the
Figure 1. Model graphs that were used to generate training data for physics-informed DEEPONETS. Green edges are used train inflow model, blue ones for inner model and red ones for outflow model.
The training data, i.e. boundary and initial conditions, are assumed to be given, as function of discrete time, in certain sensor locations, and are collected in a vector uenoor E RT Therefore, a deep operator network maps (uenssrr t, 2)) to the solution of the respective PDE on an in- dividual edge with initial and boundary conditions encoded in u sensor Our physics-informed DEEPONET does so by incorporation of residual terms that involve the woint-wise evaluation of the PDE as well as boundary and initial con- ditions. This flexibility of learning the solution of the PDE operator, i.e., the solution of the PDE for arbitrary boundary conditions veessor makes them a viable tool in our method.
vertices in VK.
To learn the parameters of our model, we minimize the objective
ppds
Piii
t2, 2,) +
Σ
Lini Caai(11) (2,
gensor .23)
Ledge (u. sensor
(t)
(9)
where 0 is the set of trainable parameters of our model, Tpdee Ponit and Tae are the respective batch sizes. By setting Go Guemne (1, == Go (uscusor x) as the output of the inflow (resp. inner and outflow) operator network, the pointwise PDE loss is defined as
To generate training data for the trift-diffusion model on the metric graph we rely on a finite volume implementation described in Section B where we fix be 1 for all edges and T = 1. We solve the PDE (1)-(3 on three kinds of graphs depicted in Figure I using this finite volume method (FVM). Initial conditions iiit as well as inflow and out- flow conditions u wnllow and quillow are obtained by sampling from a Gaussian process for all edges e E E and all ver- tices E VDD resp. These are evaluated on equidistant discretization, both in space and time. We assume that all random Gaussian processes are approximated through
AnDEEupeea sensor, t, 2;) = (HIC) (1., (!r
LYDE(",
where the DEEPONET operator is learned to satisfy the physics model. For the initial data we use
Linit (uqqusor, xx)) = =c;² (0, - ue
iiitt (x:))
2
g(x)= = Σ Th d(x - xx)
(8)
The edge loss is the only term which differs among the three different edge types. We train the inflow model based on
1=1
by using radial basis function (RBF) kernel d(r)
Ciaflow (u.ensor; t.) cdlge
exp (-1m-1// /e2) with length scale = 0.5, NGP = 512 equally distributed centers I& E [0. 1] and mi ~ NO. 1) normally distributed.
origin (tx) (1 G (44,0) - Je (t,, (e)
(4.
uu, (ungreet (tx) GG uu (t. 1)))
To accommodate for discontinuities in the initial conditions of the randomly initialized graph, we let the finite volume scheme run for a small time-interval and then take the so- lution at this time as the initial solution for the training of our model at the sensor locations init to obtain init along origin each edge. The training flux sensor measurements me and the turget are taken similarly at sensor locations grrggn and starget, resp., by evaluation of the flux boundary condition (6) if either the origin or target vertex belong to Vp. and by evaluation of the Kirchhoff-Neumant condition (4) for
while the loss for the inner model is given by
Cinner (u sensorr :ta)
L (riigin(t)) - Je (tc, (())))))
JC 1 (4, 1)))
Similar to the inflow edge loss, the corresponding outflow
4

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Physics-Inee Plysice-Informed DeepONets for drift-diffusion metric graphs
edge loss term reads
Cinde (u, semsar;
Ja(G'' (42, (1)) tz)
M (t,)
Larget U4 (L:) Go (4,11) Ja grnr (1i, 11))
In contrast to the default DEEPONET approach, which is trained using a large number of point evaluations the solution obtained using some reference numerical method, the thysics-informen approach only relies the physical model in arbitrary points well as set of reasonable initial and boundary measurements.
The model architecture in the approximation of the opera- tor net follows ( (Wang et al., 2021). In particular, we use a modified multilayer perceptron (MLP) as branch net and Fourier network with random frequencies as trunk net. We train our models with seven hidden layers and hyperbolic tangent activation function. Training of all models is con- ducted with gradient clipped Adam optimizer (Kingma & Ba, 2015) with an exponentially decaying learning rate schedule and 20 000 epochs. To investigate the influence of the expressivity of the networks, we train a small network with seven hidden layers and width 100 and large one with hidden dimension 200 for the various edge types. For each model,
Figure 2. Illustration of random GP training data: initial condition measurements u (blue), inflow measurements thp outflow measurements u.
inflow (green).
outflow (orange).
(t, x
(Worigin Utarget Winit
single precision on single NVIDIA A40 GPU with three different sets of training data: the 5K, 10K and 20K models use data generated from 5000, 10 000 and 20 000 FVM solves with random measure data, resp. Since our training graphs depicted in Figure contain in total inner edges, inflow and outflow edges, the inner model trained with 50 percent more data than the inflow and out- flow edge model. We decided keep this slight imbalance, due to the fact that inner edges appear much more often than boundary edges, especially in larger graphs.
Trunk Net
Branch Net
G (t,
We report the validation loss for each model in Table One can clearly see that the approximation quality our model improves significantly more training data are used. Furthermore, the larger model with width 200 benefits even more from larger training set and halves the validation loss when compared to the smaller network. Convergence plots of the various loss terms can be found in Section D.
H(G
GL Flow
G (0,.r) Initial
PDE
Ledge
4. Model evaluation
LPDE
Linit
Loss
After performing the training procedure discussed in the previous section, we obtain three DeEPONETmodels with different sets of parameters for inflow, outflow and inner edges, resp. We denote them by Go Go.... and G where we always use letter G, as the architecture is the same all three cases. Each operator network Gg. for = Outtow, funner fouutoww returns function evaluations of its solution in arbitrary points (t, .)) for arbitrary feasible
min
6*
Figure 3. Illustration of physics-informed DEEPONET adapted to our setting from (Wang al., 2021).
5

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Physics-Informed DeepONets for drift-diffusion metric graphs
Chein Graph G1
Double-Y Graph G2
Binomial Graph G3
Width
Data
Inflow
Inner
Outflow
5K
4.34e-03
1.42e-03
3.63e-03
100
10K
1.62e-03
8.29e-04
1.95e-03
20K
1.09e-03
5.67e-04
1.05e-03
5K
8.09e-03
1.42e-03
6.07e-03
200
10K
1.96e-03
5.50e-04
2. 2.122-03
Figure 4. Model graphs that were used to verify our methodology.
20K
6.64e-04
2.62e-04
6,30c-04
Table 1. Final validation loss after 200 epochs of training.
Width
Data
G1
G2
G3
G4
SK
5,50e-02
3.27e-02
3,97e-02
6.65e-02
100
10K
9.38e-03
1.29e-02
1.31e-02
1.11e-02
boundary and initial conditions, separately for each edge. Pursuing our goal to solve the drift-diffusion equation de- fined in (1) on a complete graph, it remains to obtain the correct input parameters on each edge such that the conti- nuity and Kirchhoff-Neumann conditions given in (5) and (4) resp.. are satisfied each vertex. Learning these un- known flow parameters is done by minimization of the loss function
20K
8.46e-03
1.03e-02
1.10e-02
1.44e-02
5K
2.87e-02
1.71e-02
2.35e-02
3.36e-02
200
10K
6.06e-03
7.66e-03
7.59e-03
6,86e-03
20K
4.68e-03
5.62e-03
5.81e-03
7.91e-03
Table 2. Absolute space-time e'--rror between solution the
FVM code compared to the output of DEEPONET averaged over 1000 runs with randomly drawn initial and boundary conditions. These are sampled as Gaussian processes (8) with Ray 468 and = 0.4.
Comupting (z) (z)= (x)==<e 2))
-t,
(10)
where the values for inflow, stored in vriggn and initial condition wnit are known and the values for the outflow, encoded in ungge have to be determined. Since arre is parameterized using the above-deseribed RBF interpolation we now learn the parameters 3 for the outflow condition. To address this challenge on the whole graph we learn the values for 33 at all nodes to enforce the PDE, the coupling and continuity conditions as well as initial and boundary (inflow plus outflow) conditions. The parametrization of this system only requires 2ng parameters for all inner edges and 14 parameters for inflow and outflow edges. With na, small the resulting learning be done in matter of seconds using a standard gradient based optimization algorithm such as Adam implemented in JAX (Bradbury et al., 2018).
where (tz, z) is defined by
Σ
(a())) (ee (1,, (li,)) Pe (I,
Vxl
[E
=EVK
e,e'EE.
continuity loss
VK
DEVI
REE.
Kirchhoff loss
where the first term ensures the continuity of the flow values at each node for all edges. The second Kirchhoff term en- sures that the conservation mass across the overall graph and all nodes. Here, the value ="(x))
(x) (f;,v) v) corresponds to the evaluation of the DEEPONET for u(z) depending on the respective edge type at time 4, Similarly, (.'(()(t, (tz, v)
The results confirm that our methodology is able to learn the solution of the drift-diffusion PDE on graphs. In the upper part of Figure we plot both the physics-informed DEEPONET and the reference solution. The error terms shown below indicate that the approximation error is small, see also Table 2 and Table 3 for detailed values. Figure 6 shows that our method is able to capture nonsmooth transi- tions at the vertices of a chain graph. Again, the solution of the DEEPONET and the reference solution are visually indistinguishable.
represents the evaluation of the flux. We here assume that the vector is approximated by a kernel interpolation using a radial basis function (RBF) kernel with fixed parameters resulting in
(t) Σ Be tx) 1=1
where n. 10 is chosen to further reduce the computa- tional complexity and la are uniformly distributed in [0, 1]. The kernel function Φ(r)= exp [|rl
5. Inverse problems
with f = 0.2. To
illustrate this in bit more detail consider the following inflow edge modeled via
The methodology developed in the previous sections is es- pecially suited for the efficient solution large scale pa- rameter identification problems on graphs, as this amounts to merely add data misfit terms to (10).
(1 origin target iiit) E Porgan Dhanga Frint
6

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Physics-Informed DeepONets for drift-diffusion on metric graphs
Width
Data
G1
G2
G3
G4
1.28e-01
7.05e-02
9.74e-02
1.59e-01
1.0
1.0
100
10K
2.15c-02
2.73e-02
3.21e-02
2.88c-02
0.8
0.8
20K
1.96c-02
2.21e-02
2.59c-02
3.80c-02
0.6
0.6
0.4
0.4
5K
5.48e-02
3.45e-02
4.79e-02
6.84e-02
0.2
0.2
200
10K
1.37c-02
1.62c-02
1.73c-02
1.59c-02
0.0
0.0
20K
1. 1.06c-02
1.20c-02
1.30c-02
1.84c-02
Table 3. Relative space-time L error between solution of the FVM code compared the output DEEPONET averaged over 1000 runs with randomly drawn initial and boundary conditions. These are sampled as Gaussian processes (8) with zip 468 and 0.4.
0.010
(01))) (o'i)ddoo an/ex
0.010
0.005
0.005
As toy application, we think of traffic network with measurement sensors located at the midpoint of each edge, We assume that they are able to measure both the density and the flux of vehicles at their respective location and as a function of time using modern sensor hardware and corresponding traffic flow estimation algorithms, see (Seo et al., 2017) for more details. We denote these time discrete measurements by Pe E " and je T and by x-meas the location of the sensor each edge. To add this information our model, we now simply extend Ceoupling by the following additional loss terms
0.000
0.000
&1
Figure 5. Upper row: Almost indistinguishable reference solution (solid) and PI DEEPONET solution (dashed) on model graph at t = 0.5 (left) and t = 1.0 (right). Lower row: Absolute difference between reference and DEEPONET solution.
1.0
(a - -Pex
(I
meus
0.5
Tlmeas
CE
0.0
+((((z) (1 mons)
Je,s mmmmssa
1.0
In Table 6, we report the various loss terms for our test graphs averaged over 100 runs. Thus, the algorithm ex- plained in Section 4 can be used tackle the inverse prob- lem without any major changes. After completing the opti- mization procedure, we automatically solved several inverse problems: Evaluating the vector we recover the unknown initial condition and also the velocities Ve on each edge, What is more, evaluating 2" (t, x) at any time in the sim- ulation interval, we also obtain access the densities on the complete graph without the need perform another forward simulation of the model.
t=0.25
0.5
1.0
t=0.50
0.5
0.0
t=077
0.5
We test our methodology on the three test graphs Gi-G3 depicted in Figure 4 as well as on the large graph shown in Figure 10 (G4) with 1034 edges and 5 inflow and outflow edges. where we choose Timas = 101. For illustration, the learned unknown initial conditions and velocities as well as the inferred solutions of a chain graph with seven edges are depicted in Figure 7 for various levels of additive measure- ment noise e = 0.1, 0.05, 0.01. We observe that we are able to recover all the essential features of the initial conditions- but also of the dynamics at later times. In particular, due to the fitting of space-tima data, the error remains roughly constant in time, at least in the eye ball norm. As for the
0.0
1=100
0.5
0.0
Figure 6. Reference solution (solid) and PI DEEPONET solution (dashed) unrolled chain graph with 7 edges over time.

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Physics-Informed DeepONets for drift-diffusion metric graphs
10
err. init
enr. vel.
||p 11--Pum Prellla
t=0.00
G
9. 9.13e-02
9.39e-02
4.42e-02
Gi
62
5.79e-02
5.32e-02
2.76e-02
<3
4.02e-02
3.05e-02
1.83e-02
0.0
61
1.86e-01
9.91e-02
6.73e-02
1.0
G2
62
1.01e-01
5.41e-02
3.79e-02
4.68c-02 63
1.98c-02
1.65e-02
t=0.50
1.08c-01
9.52c-02
4.27c-02
G3
62
6.76e-02
5.53e-02
2.54e-02
63
5.03e-02
3,32c-02
1.70c-02
0.0
61
1.11e-01
8.51c-02
4.09c-02
1.0
G4
6.29e-02
4.12e-02
2.18e-02
3.99e-02
1.58e-02
1.17e-02
t=1.00
0.5
Table 4. Absolute L -ererroors for parameter identification problem graphs depicted Figure with measurement noise 61 0.1, 0.05, = 0.01 averaged over 100 runs.
0.0
1.5
outflow, and inner edges allows the construction drift diffusion models (and similar fashion other PDEs) on complex graphs in Lego-like like way with linear complexity in the number of edges. This would allow straightforwardy the application to real traffic data, which is readily avail- able in several open databases (e.g. (Loder et al., 2019)), Adding the respective graph topology is no obstacle, while more complex traffic equations beyond the drift-diffusion model would require the adjustment the physics loss in our suggested approach, see (Piccoli & Garavello, 2006) for overview of such models.
Velocity
1.0
0.5
0.0
Figure Illustration parameter identification on chain graph with 7 edges. Reference solution (solid) and physics- informed DEEPONET solutions (dashed) for noise levels = 0.1, 0.05, 0.01. Bottom row depicts recovered edge velocities, first row corresponds to recovered initial conditions.
Software and Data
prediction of the velocity, observe that the accuracy of certain edges away from the ends of the chain have sub- stantially larger error for high noise levels than others, which we will investigate further in future works. Nevertheless, the example shows that our approach feasible for the param- eter identification problem and even suitable for possible real-time applications such traffic flow prediction.
The software and data that is necessary to reproduce the results is published on GitHub and can found under physics ieciis-inforred operator networks- for
com http://aa//eccccccccccic.oo pdes-on--mttric-graas
Impact Statement
A more systematic error analysis can found in Table 4 and Table 5, where we report the parameter identification capability of our method using the large model (width 200, 20K training data) by using space-time error measure.
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
6. Conclusion
References
We provide novel physics-informet DEEPONET architec- ture for creating surrogate model that allows the efficient solution of a drift diffusion model on possibly complex metric graph. Additionally, our model allows to solve an inverse problem for this setup at almost no additional costs. The flexibility of traing DEEPONET submodels for inflow,
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Physics-Informed DeepONets for drift-diffusion metric graphs
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Physics-Informed DeepONets for drift-diffusion metric graphs
A. Proof of Theorem 2.2
The proof based on extending ideas from (Gomes et al., 2019; Burger et al., 2020), where in- and outflow boundary conditions are treated, to metric graphs.
We will work with Sobolev spaces defined the metric graph g = (V,E, (le)eEE). We first introduce the space of square integrable functions
(E) = {w: & - I Ve E & Ue = ule e (e) (0,)))))
It is a Hilbert space with scalar product
(V,w), (ves -E Velle dx,
(Ves wele
cEE
cEE'
which induces the norm [lu][>>()) = v (u, u). Furthermore, the space (E) of functions having square integrable weak derivative is then defined by
H² {w E (E): w, E (e) and (e(v) = ww) Ve,e' E E UE Vic)
We further denote by (H' (E)Y' the dual space, i.e. the space containing all linear, bounded functionals on H (E).
Space-time dependent functions considered as time- time-dependent functions with values in function space, say u(t) X. For such functions, we introduce the norm [ = Sod [|x(t)]|x x di and use the notation 1,2 T; X) for all functions such that this norm is finite. complete analogy, we also define H (0, T; X) and the energy space
T) T.
H' (o, T; (E))).
(11)
The actual proof of Theorem 2.2 is based on the use of the formal gradient flow structure the problem, i.c. the fact that the entropy functional
E(p) 100=-//
/// E(pe log(p=) (1 Pc) log(1 Pe)) + pve x dx,
(12)
cEE
I. Lyapunov functional. For readability, we used (p.)) = pa(1 - pc) this definition, noting that the proof works completely analogous for other choices of f. upon modifying the entropy functional.
Based on this, introduce the entropy variable defined as the variational derivative of the entropy, i.e.
wp) = 8F
Σ
e(log(p=) log(1 - -Pe) + vex.
(13)
0p
CEF
such that the transformation from w to p is given by
Pe
Ve E
1
where we = w|e.
The proof is based on time-discretization and regularization strategy using these variables. To this end, let N E N be such that (0, 7]] has sub-intervals of the form
N
(0,T) = U ((k - kT], h=1
where r = n and t4 7k. Using e O.p. te (1-pe) and regularized form of (7) with w unknown
Ue Ve Pe (1 (1- pe) we for all we introduce the time-discretized
Σ/ 3EE
k-1
Pe + 15 (1 /:)) w Bupe) dr + Σ/ (Onwg Oxspe + w se) dx
cEE
+
Σ
(-uigllow (tx) (1 - --(())) + u
nutlow (tx) p ( sp()) = 0, (14)
VEVD
12

---

Physics-Informed DeepONets for drift-diffusion metric graphs
for all test functions yp (T) and given nonnegative functions u unllow added regularization term, multiplied by
outtlow
(0,T), v VD. In addition,
Our first aim is to show existence of iterates pr satisfying the (still non non-linear) equation (14). This is done using linearisation strategy and fixed- fixed-point argument. We begin by defining the set
A fp PE
Lemma A.1. Given > O and E A, the linear problem
Pe -
Σ/ 338 c
De + Do (1- Po) We Qupe) dx + Σ (O.we Duipe + we sp) dæ
CEE
(-up up
inflow
(1x+1) (1 - (1-(()) + ux mutffoo (())) p(v)) = 0, (15)
BEVD
for all ip E (E), has a unique solution w E (E) such that
| (W) S C2
(16)
where the constant C' > 0 depends only on and p
In addition, the operator S1 A + (E), which assigns some given P E A to w, being the solution to Eq. (15), is continuous and compact.
Proof. As p1 - P) 2 0 and r > 0, the existence of a unique solution w E (² is a direct consequence of the Lax-Milgram Lemma, cf. (Brezis, 2010), estimating the boundary terms as in (Burger & Pietschmann, 2016)[Theorem 3.5]. The a priori bound follows by choosing = w as a test function and applying a trace thcorem and the weighted Young inequality to the right-hand side of Eq. (15).
For the continuity of S1, consider sequence (ū,, m..) E A such that (in, mmm) - (ū, m) A and denote by w. and the respective solutions to Eq. (15). Subtracting the respective equations and choosing vp (w w.) yields the convergence + w in H' (52). This allows us to pass to the limit in the weak formulation (15).
Finally, the compactness of Sj then follows from the compactness of the embedding HL(E) - 1.2(8).
We are now in a position to the existence of iterates, i.e. solutions to (14).
Theorem A.2. For given p 1 E A with and >> 0, there exist a weak solution p² E An (E) to Eq. (14). In addition, for > 0, holds that < p² < 1.
Proof. We define the additional operator S2 (.) - A by means of
We Pla
[2ww]
e
Using that S2 is clearly continuous well the results of Lemma A.1, the operator
S S S2 o S1 A
well-defined, continuous and compact. Furthermore, it is readily observed that A is convex subset of - (S2). Thus, an application of Schauder's fixed point theorem yields the existence of fixed point p* E A associated to w/ S, (p*). By definition of S2, the H' -regularity of w directly implies p² H' (E) as well, which allows us to identify the fixed point as weak solution of Eqs. (14).
Finally, we note that as the edges are one-dimensional domains, we also have the embedding H²(E) + (E) Appealing again to the definition of S2, this implies the strict bounds < p² < 1.
Next, we use the entropy functional to show that the iterates ot are bounded, uniformly in r. This will then allow to extract converging subsequences whose limit will be the desired solution of the original problem (7).
13

---

Physics-Informed DeepONets for drift-diffusion metric graphs
Proposition A.3. Lei (0=) C CA AAHH(E) be solutions (14). Then, for any & E N, the following discrete entropy estimate holds
1 - (B(())) 1)
+ Σ /// 12,uf12 + |u dx Σ p01
-1
- 50.
(17)
EEE
EEE 12
Moreover,
0 Bizee S CC+ -
(18)
SC,
(19)
where Wy denotes the piecewise constant in time interpolation of wh and with C > 0 independent o and k.
Proof. Owing the strict upper and lower bounds provided by Theorem A.2, the logarithmic terms appearing in the
derivative of the entropy are well defined. Thus, we use the joint convexity of the energy to obtain
/
E(p²) E(p"-1) S
Σ oEE
[=(log log(1 (t)) + vex) Pe 1) dr
Using the definition of the entropy variable, Eq. (13), this results in
E(p²) - E(p -
we (oe Pe dx.
EEE
Due to the (reguuatty of w², we may use it as a test functions in Eq. (14) to get
} (E(p") E(p S -T Σ
-
(10, + 22xr))dr
3EE
Σ
(c - 1=) dx
333
Σ
inllow (tx+1) (1 p (v)) + u sulllow (tx+1) (+))
(20)
VEVD
Using the definition of w cf. Eq (13), and the fact that inflow vertices are always located at a 0 on each edge, we rewrite the inflow terms as follows
Σ HEVD
u inflow (tx+1) (1- (1 (v))w
Σ
u intlow (t++1) (b)) (log (log
(z) (log) (loo) 1 (b)))
"EVD
[-(1 -
1 (u)
Σ vEVD
infhow (tx+1)
(LI--((0) (x)) log
+ (c) -1 (c)- Σ VEVD
inflow (t++1) (- (v) + sc
(3(
We recognize the first term as a relative entropy which is non-positive, while the second term is bounded since 0- S p² similar argument implies that the outflow terms are bounded as well. Finally, using once more the definition of wk. cf. Eq (13), we further estimate the second term in (20), using the weighted Young inequality.
Σl [ 0c - (,) 12 de 2 3Eg
Σ
10
de - -2 S/¹ |uol [vell0apet dx do f/r d= p²: ve dx
P111 Pt)
3EE
pt
dx (// pe (1 p:) leoe dx
Σ
2 pt (1 -pt)
> 2 Σ
12 dx
1
Eleve.
REF
33F
14

---

Physics-Informed DeepONets for drift-diffusion metric graphs
Inserting this estimate into the entropy inequality (20) above yields
||0,1* 12 S SC+ + - - -F((()).
From Eq. (20) get
(10, + 12<e)) & SC
= - F(D²1).
For the piecewise constant time interpolation wy of w this yields, summing from &: -,..., Nr, the estimate
| w (1 + 1|w, (Iw-IB200r;;zee)) S CC +
1E(p--)--(00
Since the sum on the right-hand side is telescopic, this simplifies further to
[10 we (1 + 1| (1: S <C(+()(D)) E(p"r)),
using T T. Since S p I for e and maxxce le < co, we obtain the following uniform estimate
+ 1121000,11000001100000 S CC,
independent of 0.
Lemma A.4 (Time regularity for p.). Let (p C An (E) be the solution to the implicit Fuler approximation (Eq. (14)) and let Pr be the piecewise constant interpolation associated with (p*) Then, there holds
||d | SC,
where C 0 is independent o 0 and d, denotes the finite difference quotient
p -pp 1: k= = Ny.
[d-prlle 144)
Proof. This result follows from the regularity estimates of Proposition A.3. They allow to estimate the terms on the right hand side (14) in terms of a constant multiplied by [|e]|ance)- Thus, taking the supremum over all yields the desired (H (8)) -estimate.
Having established the priori estimates, let us now show the existence of convergent subsequences whose limits we identify weak solutions to (7).
The bounds provided by Proposition A.3 in conjunction with the Banach-Alaoglu theorem (see (Brezis, 2010)) yield the existence of subsequences and a function 0.p E (2(0,T;)3(E)) such that
O.pr - 0.p L())
where we did not relabel the subsequences. Moreover, again by the uniform bounds of Proposition A.3, we may invoke (Simon, (986))Theoren 6] such that
Pr L2
again, up to subsequences. Finally, from Lemma A.4, we have
d-p- - in (0, T; (H (8))),
15

---

Physics-Informed DeepONets for drift-diffusion metric graphs
up to a subsequence. The identification of the limits follows from standard arguments for weak convergence, see, e.g. (Crossley al., 2025)(Section 2.3].
Having collected sufficient compactness and the corresponding convergent subsequences and limits, we can now prove the main result.
Proof of Theorem 2.2. Let us revisit Eq. (14), i.e.,
T Σ/e 33E
[d-Proore + Pr. (1 - Pr.ee W )2pr,e) o) dæ + =E / (O-wr,e + wre Pr.)) dr
EEE 00
Σ .EVD
uy inflow (1)(1 - /ut (u)) ) + u7
nottlow (1)) di 0, (21)
First let note that the term premultiplied by r vanishes due estimate (19). Next, using the convergences above, can pass to the limit in the other terms of the equation get
Σ / (Orpall) + (e (l) us ((p((()) ((pp)) dx
Σ
(-u inflow (t)(1 p(t, + u. qutfow (f))(t 0,
VEVD
for any E ((,,T) x E) which dense in T;; (E)) Thus, the limit p is a weak solution to Eq. (7).
The a priori estimates follow from passing to the limit in the bounds of Proposition A.3 and Lemma A.4, using the weak lower semicontinuity of the norms, Finally, the compactness is sufficient to conclude that the weak solution satisfy the initial data.
B. Numerical solvers
Partial differential equations (PDEs) are an essential tool science and engineering, as they are typically used to model the complex physical phenomena. These equations are typically dependent on crucial system parameters that are mostly not known precisely and the formulation of the problem is written in infinite-dimensional function space setting. As result numerical discretizations of the equations are performed based. We here foeus the when a finite volume method (LeVeque, 2002) is used which was previously introduced in (Blechschmidt et al., 2022). These are popular discretization schemes as they usually work in structure preserving manner.
B.1. Finite volume scheme
To derive finite volume scheme we briefly recall our setup and start from considering differential operators defined on each edge, and we focus non-linear drift-diffusee diffusion equations
O.p. = 0n(s Q.p. (p.))) E E,
(22)
where Pe e (0, + .R4 describes, each edge, the concentration of some quantity while ve > 0 an edge-dependent velocity, and is (typically small) diffusion constant. Furthermore, f R. + R4 satisfies f(0) f1) = 0. This property ensures that solutions satisfy 0 < P. < 1 a.e. on each edge, see Theorem 2.2. By identifying each edge with an interval [0, E,), we define the flux as
Jx(x) = - == Oxpe + (p)
(23)
A typical choice for f used in the following is f(pe) = -c)
The edge set incident to a vertex v E V is denoted by En and we distinguish among Ein = {e 8: e= (ū, v) for some e V} and E.tHt = E, \ gin. The control volumes are defined as follows. To each edge e E E we associate an equidistant grid of the parameter domain
x=1//2 2//2 < < a +1/2 T.e
16

---

Physics-Informed DeepONets for drift-diffusion metric graphs
with be k+ kー and introduce the intervals E (#4-1/2 for all k: O,..., ne. We introduce the following control volumes for our finite volume method,
the interior edge intervals If:................. ................................ -1 for each E E,
the vertex patches 1 (Weesy ) U (VEEg. 1i) for each v E V.
A semi-discrete approximation of the problem (1)-6) can be expressed by the volume averages
(E(1) 1421 /
J pelt, de,
((()) = |-
/ eEEn 16
Σ
pell, do +
E/ pell, do).
1s
EEE.
for all == 1, ne 1, resp. v V. With the definition the vertex patches we strongly enforce the continuity in the graph nodes. Integrating (22) over some interval I k oe..., gives
J1z (5 02pell,x) vel( de de J,g @spell,x) de = [
" 1/2
= hh Deph = (= ve vef(pe(tt (t)))
(24)
1-12
The diffusive fluxes are approximated by central differences
0 1 (P)+1(1) (E(())
he
and for the convective fluxes we use, for stability reasons, the Lax-Friedrichs numerical flux
f(elt, #x+1/2)) de(t)= (1)))2())
"e = - (((A(() +
a - / ( -E(()),
(25)
where the convention P6 p" for E V satisfying Eoet and ph. = p² with ह E V satisfying E E The parameter > 0 is some stabilization parameter, chosen sufficiently large. At inflow and outflow vertices v E Vp we insert the boundary condition (6) into (24) and obtain
Σ (c))pe(t.v) (t,((p(t,,)))) dz())) x -u infllow (t) (1 p²) + u outllow (t) p². 3
Combining the previous investigations gives the following set of equations for each control volume I h: = 1,..., ne 1, e E &, and 1", V, respectively.
For each e E E and k = 1,..., ne 1:
he 0.pp(t)) + (t) + (t)
(h+
FR + k+ (1) =
(26a)
he
For each v E Ve:
p²(() (1) F (t)
Σ BEE
he (t)+ Σ
he
CEEn
pi(t)
(t())
Fíg (t) =
(26b)
he
17

---

Physics-Informed DeepONets for drift-diffusion metric graphs
For each influx node VB
pi(t) (1)
Σ cE
he (1)
Σ
F FT(1))
u
- (1-p -
(26e)
be
EEEN
For each outflux node E D
p((t)
P
he (t)+ Σ BEE
Σ
Fee
(t)
+ up
outllow p² =
(26d)
he
Ro-
REED
In (26b) accumulated contributions evaluated in v vanish due to the Kirchhoff-Neuann vertex conditions (4).
To solve the system of ordinary differential equations (26) for the unknowns P and p", respectively, introduce the following time-discretization. For some equidistant time grid 0 t1 <... < tae T with grid size t, bn-1.
1, ,,.., Rh,, we define the following grid functions by
(tn), =i R++1/2 F (1
We restrict the equations (26) the grid points and replace the time derivative by difference quotient, evaluate the diffusion terms in in+1 and the convective terms in in- This yields for each :,...:7.. ne the following system of equations:
For each e E & and & = 1,..., The 1:
he cm + T -PA-1 + 2pp
Pk+1
tee Pk
c-n-l
+
(27a)
he
For each E Vc:
1
c.n Pne
P1
Lel p pm TTE
fie
CEE
cEEgu
[el pm
Fe pe,n-1
Σ
pea-1 1
(27b)
Re
1
For each influx node Vin:
p1 D
[el p T8
[n| pmn-1
pen-- 2
T u
inflow (1 v,nn1).
(27c)
ba
eEEN
For each outflux node E V
ITl p.. +TE
Σ
P
[bl pmm-1
pe.n-
21
outflow
p pn
(27d)
be
ne
6EEn
The initial data are established by
c.0 = (p))
p p0 = = (po):
where TM denotes the 2 projection onto the constant functions subset M C Г. Note that this set of equations linear in the unknowns in the new time point PE k I,..., ne p: [0. 2] x + IR then reads
1, E & and nmhn, E V. The fully-discrete approximation
p(t,x) = (x) for 1 (ln tnt1)
with
f I", ('(x) - Pk enn for & E IE.
It is well-known that finite-volume schemes like (27) guarantee couple of very important properties. On the one hand, there is well established convergence theory, see e. g. (Morton et al., 1997; Lazarov et al., 1996; ten Thije Boonkkamp & Anthonissen, 2010). On the other hand, our scheme is mass-conserving and nound-preserving which we show in the following theorem. Thus, the finite volume approach is suitable of generating reference solutions used to train the DEEPONET models proposed here.
18

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Physics-Informed DeepONets for drift-diffusion metric graphs
Theorem B.1. The solution of (27), p. satisfies the following properties:
i) The scheme is mass conserving, i.e., ff = Ut uoulloo
= ffr all U E VD. then there holds
[ 0 f0 dr f dx
vn l......,
ii) Assume that f(x)) xx11 x) and in (25) choose a 1. Then, the scheme is bound-preserving, i.e., there holds
x), E [0 V [0, T,,,,
provided that S minegg he-
Proof- i) This directly follows after summing up all the equations in (27). Note that the diffusive and convective fluxes cancel out.
ii) The system (27) can be written as system of linear equations of the form
(M +T=A)Ø" Mp 1 F((""))
(28)
where M is the mass matrix and A contains the coefficients of the diffusion terms on the left-hand side of (27). The
vector 9dh contains the unknowns p". and e.n In the usual ordering of unknowns and equations the matrix M T A is strictly diagonal dominant and thus an M-matrix. The inverse possesses non-negative entries only. The right-hand side of (28) is also non-negative under the assumption pm-1 E [0, 1]. We demonstrate this for the equation (27a).
Insertion of (25) and reordering the terms yields
1 (he aT) Pk cnn- he Pk c,n-1 +7 R-1122 k+1122
(11 (1- Pk-1 - + aa) pon-l + (-(1 - + 2) cnn-! 2 0.
The non-negativity follows from Pk 0in-1 E [0 for h: = .,..., me and a = as well as T S minegg he This, together with the M-matrix property of M TE A, implies p" 2 0.
Due to f(x) = xx1 x) we may rewrite (28) in the form
(M (M+TeA)(I-P)) p²)
((pī-pー11) -(i
with some vector-valued function G. With similar arguments like before we conclude that the right-hand side is non- negative and thus, pn 2 0, which proves the upper bound. By induction the result follows for all n 1,.... ne.
B.2. DEEPONET further details
The mathematical foundation of DEEPONET is rooted in the concept of approximating operators, which are mappings between dnfinite-dimensionn function spaces. Let be such operator, which maps the input function u(x)) to an output function (v)) where x E Rd and Th are the input and output coordinates, respectively. The goal of DeepONet is to approximate G using a neural network architecture that can handle functional inputs and outputs.
In more detail, the DEEPONET architecture consists of two main ingredients: namely, the branch net and the trunk net. The branch net takes as input the discretized values of the input function w(z) at a set of predefined training points {21, P2,...,...... ., Jn }. and gives as an output set of coefficients (b), bg..... ). On the other hand, the trunk net takes as input the output coordinate y and outputs set of basis functions {41 (y), l2 (w),..... (..)..... (w)) The final output of the DEEPONET is then given by the inner product of the branch and trunk outputs written
G(u)(v) Σ (w)) 4.(),
where in this equation the coefficients (u) are obtained from the branch net, and the coefficients (y(y) are produced by the trunk net. With this we are able to approximate the operator & by learning the appropriate coefficients and basis functions from training data.
19

---

Physics-Informen DeepONets for drift-diffusion metric graphs
BRANCH NET
The branch net architecture is used for encoding the input function u(x)) into finite-dimensional representation. Given the discrete values of u(x:) at known points { .... ) the branch net processes these values through neural network architecture to produce {bi, .2...., Bp}. In summary, the branch net can be represented function B IR" R² such that:
b = ((u(xi), u(x2), w(xn)),
where b- = =bb1, b2, ... .p]" is the vector collecting all the coefficients.
TRUNK NET
The trunk net is then used for generating the basis functions that are used to construct the output function. For the output coor- dinate y, the trunk net processes y using a deep learning architecture to produce the basis functions t22))) (y)) Again, we obtain the following representation T m IRP such that:
(wy) = T()),
where t(y) = (V), (..,))......(())))
=Tp())] is the vector of basis functions.
The training of a DEEPONET involves minimizing a loss function that measures the discrepancy between the predicted output and the true output. Given a dataset input-output pairs {(us, G(u,))) the loss function defined as:
N L
/
2
((w))o))
Σ (us)))
(u) dy,
N
where the integral is taken over the domain of the output function. In practice, the integral is approximated using numerical integration techniques, such as Monte Carlo sampling or quadrature methods. The parameters of the branch and trunk nets are then optimized using gradient-base methods, such as stochastic gradient descent (SGD) or its variants, to minimize the loss function.
C. Comments
The DeepONet approach shares a lot of similarity (even equivalence) with the FNO approach as pointed out in (Kovachki et al., 2023), As a result one could apply our surrogate coupling technique with different choice of the DeepONet architecture to obtain an FNO setup and vice versa. Our methodology of coupling surrogate models based on the graph topology can be used with different operator learning methods, e.g., (physics-informat DeepONets or FNO. Also, one could use a Graph Neural Operator technique that can act in the same way as the DeepONet FNO for one edge operator thus creating surrogate operator, see(Li et al., 2020).
C.1. Discussion strong GP Prior
While we use the Gaussian (RBF) kernel in several places our main goal avoiding an inverse crime was to choose different parameters for the generation of training data (inflow/ottlow/niitaaa::: length scale / 0.5 and 512 Gaussian centers), to generate random data for simulation (inflow/outllo//nittat.. length scale é 0.4 and 468 Gaussian centers). In the inverse setting, we employ length scale of l. = 0.2 and only 10 Gaussian centers to learn the flow couplings and unknown initial conditions.
C.2. Further applications
Beyond the toy example of traffic flow considered here, at least two more applications come to mind: The first is the transport of cargo inside of biological cells that is realized by molecular motors traveling along network of one-dimensional filaments (i.e. the graph our setting). Previous work by different authors have demonstrated that, starting from an accurate microscopic model, a mean field limit produces exactly the type of drift-diffusion equations that we consider, (Bressloff Levien, 2015; Bressloff Karamched, 2016). Finally, when studying the transport in gas networks, (non-linear variants) of
20

---

Physics-Informed DeepONets for drift-diffusion metric graphs
$0.20/10
a) Inflow models
b) Inner models
Outflow models
Figure& 8. Terms of loss function in training of models with width 100 and 5K (up). 10K (middle), 20K (down) training data. Solid lines report training loss of various terms, dashed lines report validation loss. Recall that although only PDE physics loss (pde ph 1 log), houndary physies loss (bnd_ph_1oo loo) and initial condition loss (ics_log) considered in objective, all terms decrease during training. The a-axis shows the learning epoch and the y-axis the value of the corresponding loss term.
drift-diffusion equations also appear approximation to the (otherwise hyperbolic) governing equations. They go under the name ISO3 model for gas transport, (Domschke et al., 2021).
C.3. Computational complexity
For the pure simulation task, the FVM solver is typically faster than method. This is caveat of most physics-informed neural network and operator network approaches. However, our methodology shines in the inverse problem setting, were dedicated approaches are needed for each problem type. To the best of our knowledge, unfortunately, such solvers don't tt exist for our specific setting and a direct comparison is infeasible. However, we estimate the complexity involved in both approaches for the inverse problem as follows:
FVM:
O NGradiemSStpp
No (Me Nedges + Nvertices) )2)
DeepONet:
O GradiemtSips
((3 ng + 1) Nedges))
and thus a reduction from quadratic to linear complexity.
D. Loss plots
The loss function evolution over 20 000 epochs is illustrated in Figure for the model with width 100 and for width 200 in Figure 9.
E. Results on large graphs
We test our method on larger graph networks with more edges using a directed network construction varying sizes (102,
306, 1034 edges). This is easily done by providing the adjacency structure and the inflow and outflow nodes. These examples
21

---

Physics-Informed DeepONets for drift-diffusion metric graphs
Lav
200
a) Inflow models
b) Inner models
Outflow models
Figure 9. Terms of loss function in training of models with width 200 and 5K (up), 10K (middle), 20K (down) training data. Solid lines report training loss of various terms, dashed lines report validation loss. Recall that although only PDE physics loss (pde_ph_log boundary physics loss (ond.ph.log) and initial condition loss (cs))oo))aaooooooooooooo))))oor) are considered in objective, all terms decrease during training. The a-axis shows the learning epoch and the y-axis the value of the corresponding loss term.
22

---

Physics-Informet DeepONets for drift-diffusion metric graphs
Figure 10. Graph with 1034 edges, inflow nodes (green) and outflow nodes (red).
also contain multiple inflow and outflow nodes, see for example Figure 10.
The local accuracy of our model is guaranteed by the accuracy of the surrogate model, in our case the physies informed DeepONet, which makes that the PDE is resolved up the precision of the training of this model. The overall accuracy of the coupled surrogate models depends on the solution accuracy the vertices, only, which enforce for the least-squares solver. The scaling to large networks therefore depends the robustness of the least squares solver. Here, we rely on the JAX implementation of an ADAM SGD method which should scale well with increasing network size utilizing the computational power of the underlying GPU.
An exemplary simulation on the graph depicted Figure with noise 0.01 yields the following an absolute L" error of 1.67e-02 and relative 2 eтror of 4. 02. The absolute error the solution different time steps is shown in Figure 11.
An exemplary inverse problem the graph depicted in Figure 10 with noise 0.01 yields the following errors: I error of the solution whole domain 1.57e-02 (abs) and 3.53e-02 (rel); L error of the initial condition 5.06e-02 (abs) and 1. le-11 (rel): l еттог of edge velocity below 2.29c-02 (abs) and 2.34c-02 (rel). The absolute error of the solution at different time steps is shown in Figure 12.
In Table 6, we report the different loss terms after 20 000 epochs of training for our test graphs for different noise level averaged over 100 runs.
23

---

Physics-Informed DeepONets for drift-diffusion metric graphs
Differance bebween Pi DousOher and PVW solution at ffferent times, 12 emoo abo 5, 20, 12 encer 2.40
Dffference at t-0
3 Diffrrncce at t-0.5
Diffrrnnce at t=10
035
18
60
359
0.30
.333
375
635 -.00
Figure 11. Absolute difference of PI DeepONet solution and baseline solution of example simulation problem on graph with 1034 edges.
Differonce botwcon DocsOher and PVW solution at ffforort times, 12 enror also 1.500 s2, L2 oneer nol 2:15e 02
Difeernce at tn
3 ilfeeenc at tnd.5
Diffrrncc at bol1.
0.0
614
025
22
600
-0.00
-3.00
Figure 12. Absolute difference of Pl DeepONet solution and baseline solution of example inverse problem graph with 1034 edges.
24

---

Plysics-Informed DeepONets for drift-diffusion metric graphs
Noise
Total loss
KN flux
KN cont.
Meas. flux
Meas. cont.
1.96e+00
1.04e-03
1.47e-03
9.74e-01
9.85e-01
G1
4.91e-01
3.08e-04
4.41c-04
2.44c-01
2.47e-01
2.02e-02
1.17e-04
1.48e-04
9.95e-03
1.00e-02
1.95e+00
1.55e-03
1.05e-03
9.63e-01
9.80e-01
G2
4.87e-01
4.74e-04
3.43e-04
2.41e-01
2.45e-01
2.02e-02
1.57e-04
1.55e-04
9.90c-03
1.00e-02
61
1.94e+00
2.12e-03
7.85c-04
9.75e-01
9.66e-01
G3
62
4.87e-01
6.43e-04
3.11e-04
2.44e-01
2.42e-01
2.04c-02
2.20c-04
1.75c-04
1.01c-02
9.93e-03
E1
1.96++00
2.71e-03
5.29e-04
9.76e-01
9,77e-01
G4
62
4.90e-01
7.77e-04
2.09e-04
2.44e-01
2.45e-01
63
2.04e-02
2.52e-04
1.25e-04
1.01e-02
1.00e-02
Table 6. Final loss terms after 20 000 epochs of training on our test problems with measurement noise et = 0.1, e2 = 0.05, es 0.01 averaged over 100 runs. The results obtained using the model with width 200 and 20K training data.
25


Reviews:
[{"review_id": "z8xwlTXBfa", "reviewer": "Reviewer_FAxN", "summary": "The paper builds a physics-informed DeepONet setup for solving drift-diffusion PDEs on metric graphs. They train separate models for inflow, inner, and outflow edges, then stitch them together using a domain decomposition trick. Once trained, these models can be reused on any graph—kind of like Lego blocks. It works well for both simulation and inverse problems, and you get all that without retraining or extra overhead. Clean, efficient, and scalable.\n\n\n\n## update after rebuttal\n\nI have already given accept", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Experiments on larger graphs**\nWe built larger networks with more edges using a directed network construction of varying sizes (102, 306, 1034 edges) and apply our methodology to this. This is easily done by providing the adjacency structure and the inflow and outflow nodes. We also included **multiple inflow and outflow nodes** into our network. \nThus we were able to solve simulation and inverse problems (noise 0.01) for examples with more than 1000 nonlinear coupled edge-PDE models with errors:\n\ninference setting: L2 error below 8.80e-03 (abs) and 2.13e-02 (rel);\n\ninverse setting: L2 error of solution below 1.48e-02(abs) and 3.65e-02 (rel);\n\ninverse setting: L2 error of initial condition below 4.53e-02(abs) and 9.92e-02 (rel);\n\ninverse setting: l2 error of edge velocity below 2.63e-02(abs) and 2.45e-02 (rel);\n\nIn a possible revision we will include these results via tables of error measures and error plots for the larger PDE-network averaged over many runs similar to Tables 4, 5.\nFigures of the graphs and solutions can be found in the anonymous repository: https://github.com/anonymous-icml2025-234ailp/icml-2025-rev \nWe will include the error plots indicated by *difference3d.png shown in this repository in the revised version of the manuscript. \n\n**Figures**\nWe added an axis description to Figure 5 (see anonymous github), where the graph is embedded into two-dimensional space for $\\xi_1$ and $\\xi_2$ axis, which at this point is not associated with any physical space. The $z$ axis represents the function value of our model at different points in time. An updated figure can be found in the anonymous github repository.\nConsidering Figure 8, the y-axis represents the loss values of the various terms and the x-axis corresponds to the training epoch. We will give a more detailed explanation in a possible revision in the caption.\n\n**Discussion of limitations, different types of PDEs**\nIndeed we are considering directed networks, which is often natural for the cases when PDEs are posed on these. It would of course be possible to formulate certain PDEs on undirected networks and then our methodology should also apply. For different PDEs the choice of the PDE operator is crucial. A drift-diffusion problem with large diffusion coefficient is likely very easy to learn while a purely hyperbolic equation (no diffusion) with strong transport and/or nonlinearities will require more tailoring of the surrogate architecture. We will emphasize this in a possible revision but a detailed study of different PDE models will be a topic of future work.", "claims_and_evidence": "Claims in the paper are the following: \n\n1. Novel Lego-like domain decomposition approach to solve PDEs on graphs\n2. Graph-agnostic training of the edge surrogate DEEPONET model based on inner, inflow, and outflow edges\n3. Novel DEEPONET architecture enables robust model evaluation\n\nEvidence: \n\n1. There is evidence\n2. I'm not sure how the training is Graph-agnostic when experiments are only set on three particular types of graphs. \n3. Not familiar with related work, so I can't comment on the novelty of the architecture.", "methods_and_evaluation_criteria": "Authors propose L2 relative errors and plots to visualize predictions of the model. \nMy only comments related to the evaluation would be about plots. For instance, in the Figures 5 and 8. It is not clear what each axis represents. A bit better formatting would make it easier to understand.", "theoretical_claims": "The theoretical proof seems to be correct. However, I am unsure about that since I am unfamiliar with the field.", "experimental_designs_or_analyses": "I checked. The experimental setup looks solid—they use physics-informed losses, train on FVM-generated data, and validate on unseen graphs. The inverse problem setup is clean, too—just adds data-fitting terms. Results hold up even with noise. No major issues.", "relation_to_broader_scientific_literature": "Authors claim that the DEEPONET approach should be helpful for physics-informed neural networks (PINNS), which are used in many application areas such as fluid dynamics, continuum mechanics and elastodynamics, inverse problems, fractional advection-diffusion equations, stochastic advection-diffusion-reaction equations,  stochastic differential equations, and power systems. Given such a wide range of PINNS applications, authors' contributions must be pretty significant.", "essential_references_not_discussed": "I cannot comment on this as I am not familiar with the field.", "other_strengths_and_weaknesses": "## Strengths:\n\n\n1. Strong theoretical rigor—clear formulation of the PDE on metric graphs and sound use of DeepONets with physics-informed losses.\n2. Solid experimental validation—shows generalization to unseen graphs and accurate inverse modeling, even with noisy data.\n\n## Weaknesses:\n\n1. Some plots (e.g., solution comparisons) lack clarity or aren’t very informative without context.\n2. Missing ablation on model generalization to significantly different graph structures.\n3. Could use more discussion on limitations—e.g., scalability to larger, real-world networks or non-drift-diffusion PDEs.", "other_comments_or_suggestions": "1. Not sure how the time variable t was introduced right after formula (4)—it kind of just appears without explanation. A quick clarification or reminder that the PDE is time-dependent would help avoid confusion.\n2. Minor typo: “traing” should be “training” in the conclusion", "questions_for_authors": "1. How well does the method generalize to very different graph topologies than those used in training?\nThis would help assess scalability and robustness.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "tHUgmuFRgs", "reviewer": "Reviewer_zGZA", "summary": "## Summary of paper\n\nThis paper discusses how functionals can be flowed on a metric graph by learning surrogates of drift-diffusion equations. The method applies DeepONet backbone as the physical-informed dynamics surrogate model to learn how observations at inflow vertices can be pushed to the outflow vertices following a directed knowledge graph. Training of the drift-diffusion surrogate was done via small model graphs (c.f. Figure 1) to obtain inflow, outflow and inner operators. During test time, the learned models are extended to complete or complex graphs (c.f. Figure 4). To extend the learned surrogates over complex graphs, the author proposed to learn unknown outflow condition via RBF interpolation and henceforth a plug-and-play style of additional loss is minimized under which author claimed to be efficient and sufficient to generalize. Finally, the inverse problem over metric graphs can be learnt efficiently by introducing measurement loss training given trained surrogate models.\n\n## Score (ICML should be at scale 10)\n\n*    Originality: 6/10\n*    Soundness: 4/10\n*    Presentation: 6/10\n\n## Pros\n\n*    A novel operator learning idea over metric graphs.\n*    Relative time efficient computation to generalize to larger graph.\n\n## Cons\n\n*    Though self-consistent, authors only did self-comparison. The only change is the parameter of surrogates, measurement noises, and number of training samples. Different backbone such as FNOs, are ruled out. The baseline graph-based nerual networks, e.g. GNN, are not compared as well.\n*    Transfer from simple graph to complex graphs requires learning flow values at each vertices, impling $n_{\\beta}$ or $2n_{\\beta}$ for each edge. This implies as the number of edges grows, the complexity grows linearly. How to ensure the extension is robust when the graph grows in large scale? The training data is also given under a strong GP prior in which fitting RBF interpolant is somewhat using that prior knowledge. How to reconcile the correlation? \n\n## Questions\n\n*    Theorem 2.2 stated in line 116-140 lacks proof. Considering explain further in appendix rather saying combining proofs of two papers.\n*    Line 276, Figure 4 G2 is identical of Figure 1 G2. Any more complex examples rather than seen graph in training time? Multi-inflow multi-outflow would be more persuasive if considering different graph structures.\n*    Line 290, Figure 5 \"Almost indistinguishable\" is a very bold and wrongful claim. It is apparently different comparing left column and right column.\n*    What is the potential application of drift-diffusion model over the graph. The only example is a traffic  network toy example.\n*    What is the loss of violation of continuity and  Kirchoff-Neumann condition, respectively,  during test?", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Comparison to alternative operator learning frameworks**\nThank you for this remark. In fact, the DeepONet shares a lot of similarity with the FNO approach as pointed out in *Kovachki, Nikola, et al. \"Neural operator: Learning maps between function spaces with applications to PDEs.\" Journal of Machine Learning Research 24.89 (2023): 1-97* As a result one could apply our surrogate coupling technique with a different choice of the DeepONet architecture to obtain an FNO setup and vice versa. Our methodology of coupling surrogate models based on the graph topology can be used with different operator learning methods, e.g., (physics-informed) DeepONets, FNO, etc. The case for general GNNs is a little more difficult to compare as the main GNN architectures aims at learning hidden embeddings for the input data and not necessarily for solving PDEs on edges. Alternatively, there is a Graph Neural Operator technique that can act in the same way as the DeepONet or FNO for one edge operator thus creating a surrogate operator, see *Li, Zongyi, et al. \"Neural operator: Graph kernel network for partial differential equations.\" arXiv preprint arXiv:2003.03485 (2020)*. In this setup the graph for the GNN should not be confused with the graph that the PDE model is posed on, e.g. the possible street network. For GNO one uses a different (auxiliary) graph approximating the integral kernel function representing the PDE on ONE edge. As such it would certainly be possibly to learn a Graph Neural Operator for one edge and then use the problem metric graph for coupling these GNO models. We will discuss these frameworks in a possible revision.\n\n**Robustness for larger graphs**\nThis is a very interesting point. The local accuracy of our model is guaranteed by the accuracy of the surrogate model, in our case the physics informed DeepONet, which makes sure that the PDE is resolved up to the precision of the training of this model. The overall accuracy of the coupled surrogate models depends on the solution accuracy at the vertices, only, which we enforce for the least-squares solver. The scaling to large networks therefore depends on the robustness of the least squares solver and as we here rely on the JAX implementation of an ADAM SGD method this should scale well with increasing network size utilizing the computational power of the underlying GPU.\nIn the reply to Reviewer 1, we show the applicability of our approach to larger networks with 1034 nonlinear coupled PDEs.\n\n**Loss of violation of continuity and Kirchhoff-Neumann condition**\nWe will include a table illustrating the violation of the continuity and the Kirchhoff-Neumann condition averaged over multiple runs in a possible revision.\nBoth terms are defined in lines 290-295 of the manuscript. For a better comparison between problems that differ in scale, we would like to change the definition slightly and average the values over the number of inner vertices (outer sum) and the number of adjacent edges (inner sum).\nThe same applies to the squared measurement misfit terms defined in lines 356-361 of the manuscript, where a division by the number of measurements and by the number of edges seems appropriate.\nThe following examples for inverse problems on graphs with 102 and 306 edges show this\n\n| Problem size    |    Total loss  | Continuity  | Flux condition |  Measure val | Measure flux   |\n|-------------------|      -------------------|-------------------|-------------------|-------------------|-------------------|\n102 w/  averaging |  2.04E-02   | 2.70E-04 |   2.37E-04  |      9.95E-03  |  9.91E-03 \n306 w/  averaging  | 2.01E-02   | 3.33E-04  |  2.01E-04   |     9.78E-03   | 9.79E-03 \n102 w/o averaging  | 2.14E+00 |\t4.44E-02 | \t4.06E-02  |      1.03E+00 |   1.02E+00 \n306 w/o averaging  | 6.36E+00  |  1.85E-01  |  1.02E-01      |  3.04E+00   | 3.04E+00\n\n**Discussion of strong GP prior**\nWhile we use the Gaussian (RBF) kernel in several places our main goal to avoiding an inverse crime was to choose different parameters for the generation of training data (inflow/outflow/initial: length scale $\\ell=0.5$ and $512$ Gaussian centres), to generate random data for simulation (inflow/outflow/initial: length scale  $\\ell=0.4$ and $468$ Gaussian centres).\nIn the inverse setting, we employ a length scale of $\\ell=0.2$ and only $10$ Gaussian centres to learn the flow couplings and unknown initial conditions.\n\n**Further applications** Please see reply to the third reviewer.\n\n**Proof of Theorem 2.2**\nThank you for your remark, we agree and formulated the proof and would add it to the appendix (and also be happy to provide it already now via an anonymous link). The proof follows the steps: 1) Reformulation of the problem in entropy variables; 2) existence of iterates for a time-discrete regularized approximation; 3) A priori estimates via time-discrete entropy dissipation; 4) compactness and passing to limit.\n\nWe will adjust Figure 4 in a possible revision to reflect larger networks.", "claims_and_evidence": "see above", "methods_and_evaluation_criteria": "See above", "theoretical_claims": "See above", "experimental_designs_or_analyses": "see above", "relation_to_broader_scientific_literature": "see above", "essential_references_not_discussed": "see above", "other_strengths_and_weaknesses": "see above", "other_comments_or_suggestions": "see above", "questions_for_authors": "see above", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethics_expertise_needed": ["Discrimination / Bias / Fairness Concerns", "Inappropriate Potential Applications & Impact  (e.g., human rights concerns)", "Privacy and Security", "Legal Compliance (e.g., GDPR, copyright, terms of use)"], "ethical_review_concerns": "none"}}, {"review_id": "1wqBQRDRyq", "reviewer": "Reviewer_j9HY", "summary": "The paper introduces a Physics-Informed Deep Operator Network (DeepONet) approach for solving the drift-diffusion equation on metric graphs. The authors decompose the graph into different types of edge domains, each represented by a pre-trained DeepONet sub-model. These sub-models are assembled using physical coupling conditions at the nodes to form a global solution. The proposed model efficiently simulates forward propagation in complex networks and accurately identifies parameters for inverse problems. Experimental validation demonstrates that the method exhibits strong generalization and robustness across various graph structures.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Quantitative comparison of computational efficiency against traditional numerical methods**\n\nFor the pure simulation task, the FVM solver is typically faster than our method. This is a caveat of most physics-informed neural network and operator network approaches. However, our methodology shines in the inverse problem setting, were dedicated approaches are needed for each problem type when using traditional numerical methods. To the best of our knowledge, unfortunately, such solvers don't exist for our specific setting and a direct comparison is infeasible.\nHowever, we estimate the complexity involved in both approaches for the inverse problem:\n\nFVM: O(N_GradientSteps * Nt * (ne*N_edges+N_vertices)^2)\n\nOurs: O(N_GradientSteps* ((3 * n_\\beta + 1) * N_edges))\n\nand thus a reduction from quadratic to linear complexity. Here N_GradientSteps is the number of gradient steps needed, Nt the number of time-steps for the time-discretization, ne the number of grid points on an edge for the FVM scheme, n_\\beta the number of parameters for inflow and outflow edges, N_vertices and N_edges the number of vertices and edges of the metric graph.\n\n**Release of code**\nWe have prepared a repository that we make public as soon as possible. We would also be happy to share this as an anonymous repository with the reviewers if permitted.\nHowever, according to the ICML rules, we are currently allowed to share only figures in an anonymous repository, which we did, see the reply to Reviewer 1.\n\n**Extension to more complex equations and further real-world applications**\n\nWe are sorry for having been very brief on further applications of drift-diffusion equations on metric graphs. Indeed, beyond the toy example of traffic flow, at least two more applications come to mind: The first is the transport of cargo inside of biological cells that is realized by molecular motors traveling along a network of one-dimensional filaments (i.e. the graph in our setting). Previous work by different authors have demonstrated that, starting from an accurate microscopic model, a mean field limit produces exactly the type of drift-diffusion equations that we consider. Finally, when studying the transport in gas networks, (non-linear variants) of drift-diffusion equations also appear as approximation to the (otherwise hyperbolic) governing equations. They go under the name ISO3 model for gas transport. We will add references to these applications in the possible revision to the manuscript. Future work for other types of equation could include hyperbolic equations for gas transport, with no diffusion, which is currently not discussed in the manuscript.", "claims_and_evidence": "The main claims of the paper are well-supported by evidence:\n1.The proposed modular domain decomposition approach based on DeepONet is rigorously validated through theoretical insights and numerical experiments.\n2.The model generalizes well to different metric graph structures, as demonstrated by extensive numerical results. The error analysis is well-documented and supports the stated claims.\n3.The parameter identification experiments systematically analyze the impact of noise, confirming the robustness and accuracy of the proposed optimization framework.", "methods_and_evaluation_criteria": "The methods and evaluation criteria in the paper are clear and appropriate:\n1.The paper thoroughly describes the drift-diffusion equation modeling, the DeepONet training process with physical constraints, and the global optimization strategy based on node coupling conditions.\n2.The finite volume method (FVM) is used as the reference solution, which is a well-established numerical approach, ensuring a rigorous evaluation.\n3.The evaluation metrics, including spatiotemporal root mean square error (RMSE) and relative error, are widely accepted in PDE numerical approximation tasks.", "theoretical_claims": "The paper does not present rigorous theoretical proofs but primarily focuses on methodological contributions and empirical validation.", "experimental_designs_or_analyses": "The experimental design is well-structured and methodologically sound:\n1.The experiments cover multiple test graphs with varying topologies, ensuring strong generalization capabilities.\n2.The initial and boundary conditions are randomly generated using Gaussian processes, providing a diverse range of scenarios.\n3.Detailed error quantification is provided, demonstrating the model’s robustness under different noise levels.\n4.The parameter identification experiment is designed realistically using simulated sensor data, effectively evaluating performance under noisy conditions.", "relation_to_broader_scientific_literature": "The paper clearly situates its contributions within the broader literature:\n1.It builds upon existing DeepONet research (e.g., Lu et al.) and extends it to physics-informed PDE learning.\n2.The paper thoroughly discusses advancements in physics-informed neural networks (PINNs) and operator learning approaches.\n3.The key limitations of standard PINNs (e.g., the need for retraining for each new problem) are well-articulated, highlighting the advantages of the proposed method in terms of reusability and generalization.", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "Strengths:\n•\tThe paper proposes a modular, scalable, and highly generalizable computational framework for PDE solutions on complex networks.\n•\tIt introduces a unified framework for solving both forward and inverse problems efficiently.\n•\tThe experiments are extensive and well-documented, ensuring reproducibility and clarity.\nWeaknesses:\n•\tThe paper does not provide a quantitative comparison of computational efficiency against traditional numerical methods.\n•\tNo code or software repository is provided at this stage (though the authors mention future open-source release).\n•\tThe study focuses on a relatively simple nonlinear drift-diffusion model (logistic flux function), and more complex equations should be explored in future work.", "other_comments_or_suggestions": "To further strengthen the paper, I recommend including a quantitative analysis of computational costs, comparing the method’s efficiency against traditional solvers. Additionally, the authors should consider making their code and dataset publicly available to facilitate adoption and further research in this area. Finally, extending the approach to more complex nonlinear PDEs would enhance its applicability in real-world scenarios.", "questions_for_authors": "See above", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: Dear authors,

thank you for submitting your paper that discusses a novel method for solving nonlinear drift-diffusion PDEs on metric graphs using physics-informed Deep Operator Networks (DeepONets). Your research topic is very timely and an important building block in many applied research in various scientific fileds.
You approach leverages domain decomposition by training modular DeepONet (inflow, outflow, ... ) and coupling them through physical interface conditions. 

The reviewers identified many weaknesses (missing code, comparison with classical approaches and different baselines) but during the discussions their concerns have been partially resolved. 

I would like to urge you to improve your paper by improving the presentation clarity, adding the missing proof in the paper (appendix) for completeness and also include baseline comparison. It would be also great to include better discussion about scalability limits and computational trade-offs of your method compared to other SOTA approaches.

Again - congrats!

Best,
AC
Decision: Accept (poster)