Neural Inverse Source Problem
Authors: Youngsun Wi
Jayjun Lee
Miquel Oller
Nima Fazeli
Venue: CoRL 2024 (2024)
Keywords: Inverse source problem
Physics informed neural network

Abstract: Reconstructing unknown external source functions is an important perception capability for a large range of robotics domains including manipulation, aerial, and underwater robotics. In this work, we propose a Physics-Informed Neural Network (PINN) based approach for solving the inverse source problems in robotics, jointly identifying unknown source functions and the complete state of a system given partial and noisy observations.  Our approach demonstrates several advantages over prior works (Finite Element Methods (FEM) and data-driven approaches): it offers flexibility in integrating diverse constraints and boundary conditions; eliminates the need for complex discretizations (e.g., meshing); easily accommodates gradients from real measurements; and does not limit performance based on the diversity and quality of training data. We validate our method across three simulation and real-world scenarios involving up to 4th order partial differential equations (PDEs), constraints such as Signorini and Dirichlet, and various regression losses including Chamfer distance and L2 norm.

Neural Inverse Source Problems
Youngsun Wi Jayjun Lee Miquel Oller Robotics Department, University of Michigan {yswi, jayjun, oller, nfz (azz)oum @umi ch. edu https: //ww: mmintlab.... com/nisp
Nima Fazeli
Reconstructing unknown external source functions is an important per- ception capability for a large range of robotics domains including manipulation, acrial, and underwater robotics. In this work, we propose a Physics-Informen Neu- ral Network (PINN [1]) based approach for solving the inverse source problems in robotics, jointly identifying unknown source functions and the complete state of system given partial and noisy observations. Our approach demonstrates sev- eral advantages over prior works (Finite Element Methods (FEM) and data- ddtvenriven approaches): it offers flexibility in integrating diverse constraints and boundary conditions; eliminates the need for complex discretizations (e.g., meshing); easily accommodates gradients from real measurements; and does not limit performance based on the diversity and quality of training data. We validate our method across three simulation and real-world scenarios involving up to 4th order partial differ- ential equations (PDEs), constraints such Signorini and Dirichlet, and various regression losses including Chamfer distance and L2 norm.
Abstract:
Keywords: Inverse source problem, Physics informed neural network
Introduction
We are interested in differential equations g with unknown external source f:
g(x, t, Va tb (=....) f(x), : (bxx).
(1)
Our objective is to determine the external source function f(x) IR and the full mapping from spatial/temporal coordinates æ E IR* to the quantity of interest ((x)) E R" from known govern- ing equations g and partial/noisy observations. This problem, known as an inverse source prob- lem [2, 3, 4], is of significant importance in various scientific and engineering domains such as signal processing [5], fluid dynamics [6, 7], optics [8], and more.
To address these inverse source problems in robotics, common strategy is to finite element analysis integrated with optimization techniques [9, 10] physics engine [11, 12]. While these approaches focus computation efficiencies, they suffer two notable limitations. First, they rely on complex discretization and/or meshing, which affects precision and introduces significant com- plexity. Second, these approaches require unique treatment for different equations and constraints, which significantly limits their generality. Recent progress in representation learning addresses these issues via learning priors over the simulated source functions, where the data implicitly contains all the constraints and boundary conditions. For example, prior works in manipulation simulate trinsic contact, source function exerting forces deformable [13, 14, 15, 16] rigid objects [17, 18). Another example involves simulating sensor deformations as source function trans- ducing electric signals [19]. However, the performance of these approaches relies heavily on the diversity and quality of training data and does not guarantee adherence to the physics, especially with out-of-distribution inputs. Latent force models [20, 21] present an alternative approach using Gaussian processes to identify unknowns in differential equations; however, they are restricted to linear models, and it remains unclear how complex constraints could be effectively integrated.
In this paper, we propose method for solving the inverse source problem based on Physics- Informed Neural Networks [1] that enables the simultaneous inference both the source function

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and the full state given partial and noisy observations. Our method offers several advantages over prior work rohotics: compatibility with various nth-order differential equations (including ordi- nary and partial differential equations), flexibility in integrating various constraints, initial values, and boundary conditions, elimination the need for non-trivial discretizations like meshing, ease of accommodating gradients from real measurements, and does not limit performance based on the diversity training data. We open-source our codes and real-world datasets at URL.
2 Related Works
Physics Informed Machine Learning: Physics-Informed Neural Networks (PINNs) have shown promising results in solving differential equations, including ODEs, PDEs, and IDEs. PINN [1] was initially limited to solving 1st or 2nd order differential equations [22, 23, 24]. Recently, [22. 25] succeeded in solving the forward problem of 4th-order spatio-temporal differential equations by incorporating residual layers [22, 23]. Our network architecture builds upon [25, 22J's work for its capability in solving complex higher-order differential equations. In contrast, goal is to recover the source and unobserved system states from partial observations and handle constraints.
Inverse problems in PINNs primarily focus on system parameter or coefficient identification in sparse and noisy data regimes [26, 27, 28, 29, 1, 30, 31, 32, 33). Pa- rameter identification is fundamentally a different problem from source function identification; is about the internal properties and inherent dynamics of the system, whereas we are interested in ex- ternal influences that drive the system, affecting the transient and steady-state behaviors. Recently, a thermonuclear study [30] showed promising results in identifying the forcing function of simple first-order spatio-temporal differential equation without any constraints. To the best of our knowl- edge, no existing work in PINN has focused on inverse source problems with constraints, validation on high-order differential equations, and applications to robotics.
Inverse Problems in PINN:
Inverse Source Problem in Robotics: External source identifications have considerable atten- tion in the robotics from both first-principles and learning perspectives. In addition to the work cited in the Introduction, we refer the reader to Appendix. A.1.
3 Methodology
3.1. Problem Formulation: Our goal learn mapping : x + (f(x)). [(x)) that satisfies:
g( tb V V2 . .) f(x), Vo E Sr
(2)
subject to Cm (2x t V. ((r.,,,,,, 0, Vor E Qm:
where x: is a spatial/temporal coordinates we have partial/noisy access to, ((x) is the fully/partialy observable quantity of interest(c.g. deformation field, electric field, etc.), f(x) is a Lipschitz con- tinuous source function, and Qn, 2m are subsets bounded domain Q. When we approximate the mapping with neural network as Vo(ar) and (ar), this problem becomes solving for the network parameters 0, satisfying the differential equations g(x, & VaLo, ve fa(x) 0 and M constraints {C.m (x, Do, Ve
fo( JMM We cast this problem into loss:
1
I Ca, Lo, Va fo(x))² dx,
[2 Ja_ (1, Po, Ve )) -f(())) dx Σ 12 S....
(4)
where the first term represents residual losses (L,) of the differential equation (DE), and the second term represents the constraint loss. The constraint loss includes various forms of regression losses (Crey) enforcing desired output at specific coordinates (e.g., Chamfer Distance, Mean Square Error), boundary conditions (e.g.. Dirichlet, Neumann, Signorini), and equality and inequality constraints.
3.2. Modified Multi-layer Perceptron: We use modified multi-layer perceptron (MIP), which has been shown to excel at learning PINNs [22, 23J. The modified MLP's (-th layer has 2 steps:
(1) tanh (4) (1-1) b(0), 2 c. z00 (0) + d d.( (1 z z!)).
(5)

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mip
mipp
mip
mip
W000000
6
200
W f (6-
((6)
mlp
mip
fg
Figure 1: Modified MLP with L layers. Green arrows indicate fully connected layers, and red arrows indicate operations between the fully connected layer output, a, and L the number of the green box module (Eq. 5) from the input æ to the output (=(x)) (y(x), f(x)).
The first step is a fully connected layer with weight W bias b()), and tanh activation function acting on the previous layer output 2!1-1), The second step weights the output z1()) with e and from separate MLPs with tanh activation function, as in Fig. 1. Here, we propose the last fully connected layer unique to each dimension of and f. Please read Appendix. and E.4 for details.
3.3. Incorporating Inductive Bias into Network Design: Many practical robotics systems have low-frequency signals, and building network with bandwidth matching the underlying signal is essential for robustness to signal noise and successful training. We found that removing the input mappings helps the model be robust to measurement noise (Appendix-E.2), whereas prior works in A14science [25, 22] highlight the necessity the input mappings, particularly when focusing on high-frequency signals.
3.4. Non-dimeessionalization Ensuring inputs and outputs have reasonable scales is crucial for successful training [25, 34, 35). We normalize inputs to be within unit cube [-1, and each dimension of the outputs to have similar scales. For instance, if we scale inputs in [m]] unit by some value k, we scale Young's s modulus [N/m²] by g and depth measurements in [m]] unit by k.
3.5. Loss Weighting: We found that an effective training strategy is to quickly regress the partial measurements and then gradually solve for the forcing function. One way control the conver- gence speed of each loss term is by updating the loss weights every hundred epochs using each loss term's 's gradients [25]; however, this approach is expensive in time and space, especially with highly nonlinear and high-order residual losses. Instead, we recommend fixed loss weightings satisfying 50A,I|VeCr I VveyllVooreglll where VoCnog and VgL, are the gradients of the regression loss and the residual loss. Please see Appendix-F.3 for details.
Experiments
We present three inverse source problem examples in both simulation and real-world that vary in several key aspects: the order of the differential equations, the degree of observability (ranging from partial to full measurements), and the levels of noise in the data. Additionally, we demonstrate system parameter identification using the same framework in the membrane example.
4.1 Double Pendulum Problem
In this example, validate method well-known non non-linear DE system and demonstrate the robustness of our method to different levels of measurement noise. The double pendulum in Fig. is defined over a single variable, time with the quantities of interest being the cart's 's displacement q. and the angle of link1 ib and link2 ø. The black-box box external source function is the force f applied to the cart. The system comprises three differential equations without constraints:
x E H, Φ: u(x) = (q. ti d)
(6)
h q + h2p cos vi + hz0
cos cos Φ h2Ø² sin 1 - ha = f(x) E R
(7)

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h2 cos vb + hap + h 10 Q) h7 sin vb + hg@² sin( 6) = 0 hag cos + h53 cos(1) db Φ) + hag h sin(y) () higiin Φ O,
(8)
(9)
where hy through hig are scalars determined from Appendix. B.I.
Dataset: We construct three datasets without and with observation noise drawn from M(, 0.01) and (O,, 0.03) as visualized in Fig. 3. Each dataset comprises 24 examples of inverted pendulum bal- ancing with various initial conditions (see Appendix. B.2). Each example runs from 0 to 4 seconds at 50 fps, resulting in total of 200 timestamps (um: oi 0==00000000 with the initial condition.
Training: When we use the shorthand notations cb. eg. and eg for the left-hand sides of Eqs. 7, 8, and 9, respectively, the loss function becomes:
A Σ ((e) R
A2
arg min
fo)² + (cτ)² + (et)²) +
Σ 8m
((uo um + (wo -mm)²
2 + (do 2m)²)
22
12m
Here, the time domain Ω is {-|0 - S 4}, where the domain 2mm C 9 is at the observations. The parameters Å and X2 are weights for the PDE residual loss and the regression loss, respectively.
4.2 Softbody Contact Problem
In this example, we demonstrate how our method effectively handles a highly ill ill-posed inverse prob- lem given partial and noisy observations. This example deals with softbody system interacting with a rigid terrain with unknown geometry. The goal is to predict 1) a full deformed softbody geometry and 2) contact pressure, given wrist wrench w.. R and partial/noisy segmented pointcloud P
A softbody with linear elasticity interacting with a rigid environment is known as a Signorini Prob- lem. This problem is defined 3D coordinates u of an undeformed softbody x E TR3. The quantity of interest uxx) E g2 is the deformation field, and the forcing function f(x) E IR3 is contact pressure applied perpendicular to the surface. Following [36], our problem is:
a (x) n ((x) 2 Z S4 1 div (x) + eg = O on Ω (Φx) = De(x), e= (uu(x) + Vu(x)"),
R
$ x u(x) E RR
(10)
(11)
where (xx is the stress tensor, n is the normal vector at the query, is the density, g is the grav- itational constant, D the elasticity tensor, and s(x) is the strain tensor. Domain St is the entire volume of interest with the boundary wi. The domain Ω2 the four sides of the cubed sponge, where we assume no contact for simplification. Eq. 11 represent the force equilibrium the con- tact and the infinitesimal volume æ; respectively. fx) is always greater than 0 because normal tractions can only be compressive. Detailed formulations can be found in -ppendix-B.3.
We select 20 examples from [13]'s real-world dataset, where each data sample consists of a single deformed sponge's partial and noisy measurement (P m. wm.. The sponge a 46 mm cube with a Young's modulus to the Pandas Franka Emika robot with force-torque sensor (ATI-Gamma) mounted at the wrist. This dataset includes ground truth contact locations for evaluation, obtained by directly observing the contact location through a transparent acrylic plate.
Dataset:
1.1 x 10 and a Poisson's 's ratio 0.1. The sponge was attached
Training: Our model takes the wrist wrench and partial point clouds as input, and it predicts the full deformation and contact pressure. If we define the entire 3D query space as -0043, 0.43]3, all six surfaces Zg. and four sides as Qg, the loss function is defined as:
A2
Aa
arg min CD Po) +
112
+
Σ
11(()))1++
(12)
12
R2
A4
As
A
an (x))((x)
1xx)
+
Σ
ddi (x)) + +gll
Ω
|22
2
(13)

---

Gaza
Pove:
Figure 2: Real-world data collection setup using Soft Bubbles with 6 designed objects.
where CD is a single directional Chamfer Distance from Pm to estimated deformed surface point- cloud Ps = {ær + u(x)(z æ ltp and wy is a wrist wrench estimation calculated from the esti- mated contact pressure as illustrated Appendix. C.2. The third term represents an inductive bias that there
contacts the sides, and the fourth term represents Signorini constraint, where nn a (2)).. The last two terms are residual losses from Eq. 11.
4.3 Membrane-based Tactile Sensor
Here, we emphasizes the capacity of our method to solve high-order and complex differential equa- tions' inverse source problems. We show our model's ability to perform system identification with same framework and robustness to partial and noisy observations, unlike the prior works [9, 10].
The classical theory of Von Karman [37] provides a 4th-order PDE model of thin elastic plate system subjected to large deflections under external forces. This problem is defined in 2D in-plane coordinates and 3.. The quantity of interest is the 3D deformation [u, w| corresponding to the undeformed in-plane coordinates the membrane. The forcing function f(x:) E TR represents the contact pressure applied perpendicular to the surface normal. The governing equation and the boundary conditions of the membrane sensor are given by:
2 0B (oa)(a)) da p- pp=((x) E IR (14)
Ow
x = (a, E ER
T x - (u, w) R DA
W 0, u 0, f(x) on 2s
(15)
where D Et /12/112111111 2²) a constant of flexural rigidity consisting of Young's modulus E and 8 Poisson's 's ratio μ, t is the plate thickness, the biharmonic operator 2 is Dar + Øgt + 2 Gaß is the shear stress, deflection, and p is the traverse load per area from air pressure and gravity. The membrane is subject to Dirichlet boundary condition in the domain 2g Please refer Appendix B.4 for further details.
IR2 represents the in-plane displacement, denotes the out-of-plane
Dataset: Our dataset D = (Pm. consists of partial and noisy point clouds P, and internal air pressures from the pressure sensor Pm- We recorded 24 data samples from different geometries with 4 interactions per object, as shown Fig. 2. For each interaction, the Soft Bubble randomly rotates in yaw [-90%, 90%] and lowers until the contact force reaches 8 11 N. We use Kuka arm equipped with an ATI-gamma F/T sensor mounted the wrist to move the Soft Bubble. The reaction wrench measurement from the F/T sensors is used only for evaluation purposes of our contact pressure estimation. The Soft Bubble's membrane has thickness of / 0.45 mm and an ellipse with major axis a = 0.06 m and minor axis b b= 0.04 m.
System Identification: Our network can also perform system identifications via optimizing for unknown parameters. When use inflated sensor pointcloud without contact, the sourcing term is all zero fe = 0 and Young's 's modulus is the only unknown parameter. The loss function becomes
A2
A
arg min CD m. Pa) + E.0
gE
2
|| 12 f8).
(16)
Ω
where gF,8 a the left hand side of Eq. 14, Ω (a, B, w) ())² 1 Ωo is the boundary of Ω. CD is one-directional Chamfer Distance from noisy partial pointcloud measurement P... to our predicted full pointcloud Pa (a, 8, 0) + (ue, wo))(a, (a) 8) Ω}.
5

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Without Noise
Noise (0, 0.01)
Nosse (0, 0.03)
Observtam
link2
Tinkel
Rucke Toaa Extaddoo
cart
Figure 3: The left panel shows our double pendulum system with positive displacement to the right and positive angles in the counterclockwise direction. In the right panel, we include results without noise and with noise (0, 0.01) and ~ (0, 0.03) in observations. The scatter plots in the first row depict Jm (red), the (bluc), and om (green), overlaid with the estimated states go (orange), th (bluc), and Φe (green) represented by lincs. The second row shows the estimated force fo (orange) and the ground truth fn (bluc).
Full State
Source Function
Ll Error
q
f
No noise
2.117e-3
1.587e-3
7.735c-4
3.825e-1
10 0.01)
2.505e-3
2.808e-3
1.528c-3
1.640e-3
(0, 0.03)
4.683e-3
3.945e-3
5.632e-3
1.262e-2
Table 1: The LI norm error of the double pendulum's 's full state and source function estimation results with different Gaussian noises. q and are in meters, and d and are in radians.
Training:
Given pointcloud under contact P m. the loss function for contact pressure estimation is
arg min CD((P m. Ps)
+
A2 ?
Σ [g - fo)2
A
(w)
+ ||u 2 + +())
(17)
12
S2
The only difference between Eq. 16 non zero So and that the Young's modulus is not updating.
Results
5.1 Double Pendulum Problem
Metrics: We use the L1 error to assess the full states and the sourcing function estimations at unseen time queries with x4 dense samplings.
Result:The results in Tab. and Fig. demonstrate that our model can almost perfectly predict the ground truth for both the full states and the source function when there is no measurement noise. As noise is introduced, our model's accuracy gradually decreases. This trend is more pronounced in the source function estimation because accurately determining n-th order derivatives from noisy data becomes significantly more challenging with noise. Nonetheless, average force errors of 1.262e-2 [N] from noise (0, 0.03) are still a very small fraction of the scale of the ground truth (ranging from -0.3 0.3 [N]) as visualized Fig.
5.2 Softbody Contact Problem
Metrics: We consider point to be in contact when the estimated contact pressure exceeds 1,500 Pa, found to be optimal through grid search. The average Chamfer Distance between the ground truth and the estimated contact patch in our approach is 45 mm? whereas the baseline achieves 32 mm?
Baseline: Our SOTA baseline [13] tackles this ill-posed problem by training a neural network on dataset comprising a total of 3,000 sponges-environment interactions. One-third of the data includes sponge-box interactions, similar to their real-worde dataset's 's sponges-tables interactions. They uti- lize Isaac Gym's softbody simulation [38], based finite element method (FEM), which is non-

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A
B
C
D
E
Figure 4: A) Partial pointcloud measurement (green), B) reconstructed deformed geometry where the color represents predicted contact pressure ranging from to 4,000Pa, C)) reconstructed deformed geometry overlaid with ground truth contact (blue) and partial observation (green), D) estimated contact location (red) overlaid with the ground truth (bluc), E) bascline's 's contact location (yellow) overlaid with the ground truth (bluc).
differentiable and can only address forward problems. The baseline [13] takes wrist wrench, partial point clouds, and trial code inputs, and produces signed distances and binary contact as outputs.
Result: We demonstrate that our results are both qualitatively and quantitatively comparable to the SOTA baseline [13] (Fig. 4). This is impressive considering that our model had to infer from an infinite number of contact possibilities from partial and noisy measurements and physics equations, whereas the baseline [13] relies strong priors obtained from the data it was trained on.
5.3 Membrane-based Tactile Sensor
System Identification: We used v 0.5 like typical rubber [9] and p 103, 320Pa from the pressure sensor measurement. The resulting Young's modulus is E' 341, 260Pa.
Metrics: We evaluate the predicted contact pressure with 1) contact patch and 2) net contact force estimation. For contact patch evaluations, we use Intersection over Union (IoU) and bidirectional Chamfer Distance (CD). We classify a point in contact if the predicted contact pressure satisfics fa(x) > max(2500, yer entire bubble surface area [9]. Here, the ground truth is the intersection of the Soft Bubble pointcloud and the object mesh using known transformations. For contact force evaluations, we use L2 norm between the estimated wrench at the wrist (Fig. following Appendix. C.2 and the ground truth.
2 Jn fad] where T&a Jor fad A| is the average contact pressure across the
Baselines: Our SOTA baseline [9] is Finite-clement (FE) based approach, highly specialized in the Soft Bubble's 3D contact force and contact location estimation. Unlike our approach, [9] requires correspondence tracking between mesh before and after contact, special system identification pro- cess requiring a total of 205 data points, and a result refinement step for handling partial and noisy observations using a convex optimization (CVXPY). We utilized their finest mesh resolution with 749 vertices along with their best-identified model calibration results.
Results: Fig. 5 shows examples of bubble-object interactions and the resulting contact pressure estimation from our model. Tab. 2 indicates that our method excels at contact patch predictions for all shapes except for Y, producing 15% higher IoU and 22.2% smaller chamfer distances when compared to the baseline. Fig. 6 shows visualizations of both our and the baseline's contact patch estimations, highlighting that our predicted contact masks are much smoother, less noisy, and more precise at following the shapes of the objects. Although the predicted contact pressures outper- formed results contact patch estimation, the predicted contact force produced a 1.083 N higher contact force L2 error, which is about 10.8% of the average contact force scale, 9.99 N.
7

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B
C
A
Figure 5: A) Visualization real-works Soft Bubble and object (pink) interactions. The Soft Bubble is from our model's reconstruction, and the color indicates our model's contact pressure predictions. B) Real-world Soft Bubble point cloud measurement with holes and occlusions. The color represents heights from the x-y plane. The dotted line indicates the actual bubble dimension. Ground truth contact mask prediction (navy). D) Estimated contact pressure overlaid with the ground truth contact location indicated in outlines (blue).
(Peng est al.) (Ous)
Figure 6: Contact location estimation hetween ours and SOTA baselines. Yellow region is the overlap hetween the ground truth and the estimated contact patch, blue is false negative, and green is false positive.
Object Geometry
Metrics
Average
Flower
Y
Cylinder
Cross
90 Curve
IoU
0.604
0.376
0.541
0.647
0.680
0.597
0,574
Ours
CD
1.000
2.191
1.120
0.903
0,635
0,911
1.127
Force (N) 4
1.900
2.056
1.741
2.758
1,350
1.514
1.886
IoU
0.557
0.334
0.581
0.553
0,431
0,511
0.494
[9]
CD
1.026
2.357
0.933
1.083
1.750
1.514
1.444
Force [N] +
0.678
0.706
0.357
1.642
0.873
0.563
0.803
Table 2: Soft Bubble's contact patch and contact force estimation six different shapes approach and a FEM-based baseline [9]. Force is norm of contact force error.
Future Works and Limitations
While our approach has shown effective in solving inverse source problems some real-world robotics applications, we have yet to explore real-time inference capacity and representing multiple DE solutions with single networks. Interesting future work is parameterize multiple DE solutions, similar methods used in [13, 39, 16), and integrate the latent space inference for potential real- time applications. Additionally, while our method relies the known governing equations of the system, an exciting direction involves replacing these known differential equations with surrogate models identified from data [40, 41, 42). This could further enhance the applicability of our method towards unidentified systems.

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Acknowledgments
This work is supported by the NSF CAREER #2337870 and NRI #2220876 grants. The views expressed here only of the authors and do not reflect that of the NSF. We thank the members of the Manipulation and Machine Intelligence (MMINT) Lab for their support and feedback. We also grateful to the reviewers for the excellent comments.
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Examples
Initial Values
-0.03
-0.02
-0.01
0.01
0.02
0.03
-0.03
-0.02
-0.01
0.01
0.02
0.03
-0.03
-0.02
-0.01
0.01
0.02
0.03
-0.03
-0.02
-0.01
0.01
0.02
0.03
Table 3: We perform 24 examples with offsets for each value of the parameters w. o, vh and d. In other words, one parameter has an initial offset, then the offsets for the other parameters were set to zero.
A Related Works
A.1 Unknown External Source Identifications in Robotics
In the field of robotics, there has been growing focus on identifying unknown sources that influ- ence system behavior. Particularly in the manipulation domain, the system inputs can originate from two primary sources: the robot itself and the environment. Inputs from the robot are typically mea- surable through various onboard sensors, such as force/torque (F/T) sensors or visio-tactile sensors. In contrast, inputs from the environment are not directly measurable, making their identification a crucial area of research. significant topic in this research is the inference or identification con- tact formations as unknown system inputs. For example, studies such as [13, 14, 16] have focused on identifying contact locations of elastic tools (e.g., sponges, spatulas) using partial point clouds and wrench data. Other studies, like those by [17, 43, 44, 18), have investigated extrinsic contact identification using visio-tactile measurements. In other domains of robotics, unknown system in- puts often take the form of external disturbances. For instance, in underwater and aerial robotics, estimating current or wind velocity is essential for achieving optimal robot motion [45, 46].
Additionally, across various robotic applications, identifying the actual input sensor (such as force or deformation) from the transduced signals (like electric signals or depth images) is another form of an inverse source problem. For instance, [19] focused estimating tactile sensor defor- mations from electric signals, while [9, 10] studied contact pressure estimation using depth camera outputs from sensors. It's important to note that these previous works utilize either data-driven approaches [13, 14, 16, 17, 43, 44, 18, 19] or traditional methods often based on discretizations [9, 10, 45, 46). These methods have certain limitations, which are discussed in Sec. 1.
B Experiment Details
B.1 Double Pendulum Parameters
The parameters used in Eqs. 7, 8, and are as follows:
h1 m. m2 m3, h2 (mm mę), hs m2l2, h4 [7(m1 + m2).
(18)
hg 122m,, ho 12. hn g(m m2), hs m2l2,
(19)
where m is the mass of the cart, m2 is the mass of the first link, my is the mass of the second link, 4 is the length of the first link, la is the length of the second link, and is the gravitational constant (9.81 m/s²). The cart's mass is mo 0.05 kg. and the lengths of the two links are 4 = 122 0.5 m with masses
= 0.005 kg.
B.2 Double Pendulum Initial Values
We give angle offsets in d range from [-0.3, 0.3], and initial angular velocities in d range from 0033, 0.3). Our 24 example's 's initial values are in Tab. 3.
13

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B.3 Softbody Contact Problem
Here, we show detailed formulation used for Eq. 13 implementation. First, 3x3 tensors (ω)) and El ((x)
are
azy
azz
Day
yy
Oyz
and
(20)
yy
aaa_
Bus
1 Ubi +
Outy Ux
0u
BM
Exx
Ewy
Ezs
Oar
By
02
5 (Vu+ Vu²)=
Ouu 2
0ts. By
Du 0z
Ery
Eyg
Eyz
By
By
Erz
Eyz
Du +
+
duL
thu
8y
(21)
Linear stress and strain tensor relationship (ax) D aaz) was in practice implemented following Voigt notation for computation efficiency as follows:
0
0
Caz
yy
V
V
Cyu
Ozy
0zs
(1+ 11- 2v)
E
0
0
0
1-2u
2z
0
Eusy
622
(22)
Ozz
0
1
2M
Eun
0yz
Eys
Moreover, the divergence is formulated as
0dzy +
0g
dx
dy
0z
diva
V
0a
dd + 82
00
(23)
Ox
dy
0a
ddus
00
0g
0s
B.4
Von Kalman Theory
Following [47, 48), we get the following equations:
DA w + No Ow
@2w
+ N1 Ow
Oaz 2N..
= P P
(24)
do03
0:32
Na = C + (Egg), N₂ = C(Ess + (VEa), No = C(1 - v) Eas,
(25)
Jua
Ow
Oug
1 Ow 2 Eag
Jun, du.
Ow ou
Eaa
Egg-
(26)
Da
2 da
0g
2 03
Da aB da 03
t
Et
D 12
C
(27)
22)
v2
B.5 Design Choices of 3D Printed Objects
The objects we selected mimic real-world items, shown in Fig. For example, our cylinder resembles cans from the YCB object (e.g.. Pringles can, tuna can). Our dataset was collected in real-world conditions, with 3D- 3D-printed shapes specifically designed for precise quantitative analysis of contact location estimations. Using real-world objects (e.g., YCB dataset) for evaluations poses a challenge because obtaining their precise world poses is often difficult. This difficulty stems from object pose tracking methods, such as AprilTags, which are prone to errors caused by camera distortions, depth noise, and occlusions. In contrast, our 3D-printed objects are bolted to the table with known world locations, allowing us to access their precise positions with less than mm of error. summary, when using general 3D shapes in real-world scenarios, our model will still make predictions, but there will be significant degradation in evaluation metries due to the lack of access to ground truth.
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Cylinder
90 curve
Master Chac
Camporll's
Tommatt
Figure 7: Our 3D 3D-printed objects are analogous to parts of common real-world items. For example, our cylinder resembles the cans from the YCB object set, the 90-degree curve represents the handles mug and lid well as locks, and O-shape similar to handle the YCB scissors.
Experiment
q
+
L
hf
Epochs
Batch size
Double Pendulum
3
256
100,001
201
Softbody
4
256
8.001
5,000
Membrane
2
4
256
30,000
3,000
Table 4: Specifications of training and network ppper-paramtters is the network output dimension, L is the number of layers, and hf is the hidden features. For all approaches, can fit all data points in single batch.
the input dimension, q +
C
Result Details
C.1 Training Details and Hyper Parameters
We train model NVIDIA RTX A6000 GPU. For all experiments, Adam Optimizer with learning rate le-3 with decay rate 0.9 per every 5,000 iteration steps. Tab. 4 shows experic details.
C.2 Membrane Contact Force Estimation
Given a of query points (a, 3), perform Delaunay triangulation [49] to generate 2D tri- angle meshes. Using deformation field predictions from network, displace the vertices and calculate the deformed triangle area described in line 6 of Alg. 1. Then, calculate the average contact pressure force exerted at each triangle following line 7 of Alg. 1. The total wrench is the summation of the forces from all triangles as described in
line 8 of Alg. 1.
C.3 Membrane Baseline [9] Parameters
For the FEM baseline method [9], use the same optimized values for the parameters: force penalty kioee 0.3322 displacement penalty kaisp 537592 m and membrane--hicknes. normalized Young's 's modulus F = 856 Pa, as reported in [9] and its associated codebase.
15

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Algorithm Contact Force Estimation
Input: queries an array of query points (a, 3) on deflated bubble ellipse (0.06, 0.04) array of 3D deformation [u, w| at (a,, () from PINN
Input: deformation
Input: pressure an array of pressure value p at (a, (3) from PINN
contact -force <-
2: for A in Delaunay [riqueelatioon(qqerree)) do
3: (ax, .3) + (ueries(A)
4: [u, w] <- deformation A)
5:
uu w] area f- ( v1) (0s
U1. 22, 0g t- a 3,
Deformed vertices of triangle A
6:
v1
7: forcet- meam (pressuee A)) ) x area
8:
contact force contact force + force
9:
end for return force
10:
t= 0
40
Source Function Estimation
1H
0.25F
10V
60
658
335
350
036
340 320
1.5
Figure 8:
Left panel shows the RLC circuit diagram and the right panel is the source function estimation (orange) overlaid with the ground truth (blue).
D More Examples
D.1
RLC Circuit
We show that our approach can be applied even when the right-hand side of the equation consists of the source function and its derivatives:
g D th V......) 91 V[(x),..
$ : ((x).
(28)
The RLC circuit Fig. is described by the equation:
8 + 7 10c = f 65,
(29)
where is the voltage at the capacitor and f is the black box source function. For training the dataset, we simulated the RLC circuit with the ground truth source function 6e- resulting in voltage measurements Umn and generated Ik measurements. The source function estimation from training produces only 5.450 10-3 absolute mean error, aligning well with the ground truth visualized in Fig. 8.
44
3 - + fe- 9e We simulated the circuit for 2 seconds
D.2 Membrane with Noise
In this section, we evaluate our approach and FEM against observation noise, as shown in Fig. 11 and Tab. 5. Specifically, we added Gaussian noise with zero mean and standard deviations of 0.0001, 0.001, 0.002, and 0.003 [m] to depth measurements. Our method demonstrates robustness up to standard deviation of 0.002 [m], showing only a 12% drop in accuracy, whereas the FEM-based approach and the baseline [9] show performance decreases of 36% and 69%, respectively. This is an impressive result because the baseline [9] explicitly accounts for observation noise via group
16

---

Without FL
Input mapping
Ours
Figure 9: Ablation studies without the final fully-connected layers per each output dimension (Without FI.).
with sinusoidal input mapping (Input mapping), and our architecture (Ours).
Without FL
Ours
Figure 10: Ablation studies on with (ours) and without the final fully-connected layer for contact patch esti- mation tasks.
lasso optimization problem, while our approach leverages the inherent noise regularization ability of neural networks without task-specific noise removal treatments.
Noise
Raw Obs.
0.0001
0.001
0.002
0.003
Ours
0.710
0.703
0.696
0.624
0.240
FEM
0.252
0,251
0.208
0.160
0.120
Baseline [9]
0.454
0.211
0.9
0.138
0.057
Table 5: IoU of contact patch detections given depth measurements (Raw Obs.) with Gaussian
noise having standard deviations 0.0001, 0.001, 0.002, and 0.003 [m].
17

---

Raw Depth 0.0001
0.001
0.002
0.003
Ours
FEM
Peng et.al.
1600
Figure 11: Comparison of contact pressure estimation between our method, FEM, and the baseline [9]. Each column shows the contact pressure distribution given raw depth measurements and with Gaussian noise having standard deviations of 0.0001, 0.001, 0.002, and 0.003 [m]. The baseline includes an internal 'regularization' process to correct point cloud measurement noise. The 'FEM' variant is modification of [9], which omits the 'regularization' phase and relies solely on FEM for solving the inverse problem. The light blue regions in the FEM and [9] results indicate contact locations found via thresholding.
E
Ablation Studies
E.1
Final Layer
We propose to add a fully connected layer for each output dimension, component absent in the architecture of prior works [25, 22]. We demonstrate that this additional layer brings significant improvements, particularly in source function estimation. Qualitatively, in Fig. 9's contact pressure estimation results, our approach with the final layer shows sharper and more accurate contact patch estimations, closely matching the ground truth. In contrast, the contact pressures estimated by [25, 22] are widely dispersed and fail capture the shapes of the objects in contact. Furthermore, quantitatively, for three randomly selected examples, we observe
increase in Intersection over Union (IoU) from 0.498 to 0.661 for the flower, from 0.504 to 0.780 for the cylinder, and from 0.298 to 0.667 for the 90-degree curve, respectively, shown in Fig. 9 and Fig. 10.
E.2 Sinusoidal Input Mapping
In this section, demonstrate how the sinusoidal input mapping[25] proposed in prior works makes model susceptible to measurement noise, which is inevitable in real-world experiments. Using
18

---

With Input Mapping
Ours
Figure 12: Ablation studies on with sinusoidal input mapping and without the input mapping (ours). Here the noise (O,, 1) was added the measurements and the estimated force functions are sub- stantially more noisier than ours without the input embedding.
L1
//
f
Input Mapping Ours
3.654e-3
4. 4.139e-3
2.405e-3
3.202e-2
3.218e-3
3.497e-3
1.880e-3
1.63e-2
Table 6: Ablation studies on input mapping given measurement noise A (00 0.01). We report the LI norms of the full state estimation and forcing function estimation. q and f are in meters, and si and é are in radians.
the same double pendulum system described in Sec. 4.1 with measurement noise ((0, 1), compare the results with and without the sinusoidal input mapping of [25]. We utilize four examples with initial offsets of -0.03 in vi db, vi, é, respectively. Fig. 12 qualitatively demonstrates that our model without the imput embedding exhibits superior robustness in force function estimation under input noise. Quantitatively, Tab. indicates that using input mapping produces 47% more errors in force term estimation compared to not using it (ours).
Moreover, demonstrate how sinusoidal input mapping degrades the forcing term estimation sults in contact location estimation performance. Our proposed architecture, without the input map- ping, increased the IoU score from 0.538 to 0.661, 0.606 to 0. 0.780, and 0.399 to 0.667, as visualized in Fig. 9. This represents a performance increase of 22%. 28%, and 67%, respectively.
E.3 Loss Weighting
Conventionally, shown in [22, 25], prior works have insisted on keeping the norm of each weighted loss's gradients equal to each other for fixed number of epochs. However, these weight- ing methods, which have been classically used in Al4science, are not only 1) expensive in time and space, especially with high-order and non-linear differential equations, but also 2) ineffective for the Inverse Source Problem. This is because the solution to inverse source problems tends to remain near the randomly initialized forcing term while quickly minimizing the DE residual loss. We propose new approach that involves using a fixed loss term that produces higher loss gradi- ents (by factor of 50) on regression losses that use partial or noisy observations the beginning of the training. This approach facilitates the network in first converging to the left side of the DE using observations, that the foreing term converges to the identified left-side value via the slowly converging DE residual loss. We show the effectiveness of our loss weighting in Fig. 13 and Tab. 7.
The results, for instance, show that the loss weighting with dr_ = 100 shows 90 % more accuracy
Area
19

---

500
100
50
10
1 (baseline)
Membrane
Softbody
Figure 13: Visualization of the forcing term in the membrane (first row) and soft body (second row) examples with different loss weightings, ranging from A 500 (second column) 5 (sixth column). The first Xeep
column of the first shows the ground truth interaction between the membrane and the 3D 3D-printed objects (pink), with the point cloud observation visualized green. The second sixth columns display pressure estimation results, with the ground truth contact patch outlined in blue. first column of the second row presents the ground truth contact estimation (hlue) overlaid with a partially noisy point cloud observation (green). The color scale of the contact pressure distributions matches that of Fig.4 and Fig.5. We ohserve that contact location detection for both cases significantly deteriorates qualitatively when Ar_. < 50. A quantitative analysis is provided in Tab. 7.
contact
Sreg
500
100
50
10
1 (baseline)
Aveg
Membrane IoU t
0.630
0.592
0.603
0.480
0.310
Softbody CD
46.275
25.587
14.912
55.050
55.702
Table 7: Quantitative evaluations of results of different loss weighting in Fig. 13. where the evalua- tion metric for the soft body IoU and for the membrane Chamfer Distance (CD).
in loU for the membrane example and 54 % less errors in softbody example compare to A suggested in the prior works [22, 25).
Areg
E.4
Architecture
We evaluate our method's ability to handle high-order differential equations through the membrane equation's 's forward problem. This example involves a simplified membrane system described by the equation:
DA = P
(30)
with Dirichlet boundary conditions. The main differences between Eq. 30 and Eq. are that Eq. 30 does account for membrane stress in the force balance equation, which holds true only the deformation is small enough and in-plane deformation no contact, i.e., f 0. This simplified system differs from the actual system, resulting in larger deformations. However, removing the stress term makes solution derivation much easier, which is useful for quantitative analysis on deformation predictions. Here, we solve forward problem given non-updating system parameters (E, v.p, D. t) as follows:
O. Moreover, Eq. 30 assumes there is
arg min
DA' wg - -P} + A Σ
(llwol + ||
(31)
f
80
20

---

PINN
Causal PINN
Ours
Tap-danp
<lepdann
Ene
Tour
Figure 14: Ablation studies on architectures for solving forward membrane equations. Black pointclouds are the ground truth from the analytical solution, and the colored pointclouds are from each model. The color represents absolute error in deformation predictions.
where À is weighting term. For further simplification, we assume that the closed domain has fixed radius r:
((a, ((a,3)la² 82 S =²
Ground Truth: The solution of the above system is the ground truth vertical deformation w given by oD (a² 32 r2)2.
System Parameters: The membrane thickness is f = 3 x 10 [m]] Young's 's modulus is E' = 100, Poisson's 's ratio is v = 0.5, internal pressure is 0.4 psi = 2.757 x 10 [N/m²), and D is Et//12( 112)).)). Also, we assume that the geometry is a sphere for easier ground truth computation v²).
with radius 0.04[m], which the length of the bubble sensor's semi-minor axis.
Evaluation: We pick optimal network parameters 0 producing the minimal training loss shown in Eq. 31. We use 10,000 unseen test queries for the evaluations, randomly sampled in the domain. We use Wg w| for the quantitative evaluations.
PINN [32]: The original PINN recommends sampling different query points for every iteration. For every iteration,
11,000 query points, where 10k points are from 0.04 domain and 1, 1,000 points are from the boundary. [32] recommends the following parameters for training: the number of layers is 4, hidden dimension is 20, and activation is tanh. Also, we found these parameters helped convergence: epoch=20k, Adam optimizer with learning rate le-3, and loss weighting \ = 1,000.
CausalPINN [22]: CausalPINN recommends the same training data sampling method as the origi- nal PINN. CausalPINN has very similar network architecture to ours except for the input mapping. We follow CausalPINN's 2D Navier-Stokes example's Fourier input mapping 7 given by:
7(, B) [1, sin(ak,w,), cos(a, w..), sin (8ki wyy, cos (Bk,
(32)
sin(ak,w,,) sin sin wy)) (akiig wy) cok( cos(akk, w, (ihh, wy,), cos(ak, wy cos (Bkk
(33)
(34)
where Wa wy 2T and k2 ky 1. We found the following parameters to be optimal for training: number of layers 5,, hidden dimension 256, activation = tanh, epoch=20,000, Adam optimizer with learning rate 1e-3, and loss weighting Y1 1,000.
Results: Fig. shows the results of the membrane equation's forward problem with different ar- chitectures. The maximum deflection of the ground truth 36.7 mm, whereas the original PINN's maximum deflection is ~ mm, Causal PINN's is 6.5mm, and ours is 36.5mm. Ours only produced 0.2mm of emror whereas the other architectures almost completely failed to solve for the deforma- tions given the air pressure of the membrane system.
21


Reviews:
[{"review_id": "Psqs36YWUJ", "reviewer": "Reviewer_FvMb", "summary": "", "questions": "", "limitations": "", "rating": 2, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": 2, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "This work proposes to solve inverse source problem of PDE for robotics applicaiton, solving source function and state mapping at the same time. The proposed method is based on PINN and many examples are given to show the results, including one real-robot example.", "summary_of_recommendation": "Incremental contribution, lack of insight for PINNs in robotics, poor oganization.", "technical_quality": 3, "clarity_of_presentation": 2, "potential_impact": 2, "robotics_focus": 4, "quality_of_limitations": 2, "questions_for_rebuttal": "See weakness above.", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pvrmBjMEpO", "reviewer": "Reviewer_dC7P", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": 3, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "This paper proposes to use PINN to solve the inverse source problem in robotics", "summary_of_recommendation": "The paper introduces a new tool to the inverse problem in robotics. It can have a good impact for the following methods.", "technical_quality": 3, "clarity_of_presentation": 4, "potential_impact": 3, "robotics_focus": 4, "quality_of_limitations": 3, "questions_for_rebuttal": "1. Could you explore more on FEM results? Is it due to the noisy data? Can PINN outperform FEM in simulated experiments without noise?", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pNeIHWtAT0", "reviewer": "Reviewer_D4sA", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": 4, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "In this paper, the authors introduce a novel way to solve inverse source problems by exploiting Physics Informed Neural Networks with a modified Loss function. This structure allows to reconstruct the external source function based on partial and noise observations of the system. The proposed method is evaluated on simulations and real-world datasets. In comparison to existing approaches, it shows superior performance regarding the accuracy of the predictions.", "summary_of_recommendation": "Simple but effective approach to learn the source function of an ODE or PDE with PINNs with convincing numerical evaluations.", "technical_quality": 4, "clarity_of_presentation": 4, "potential_impact": 3, "robotics_focus": 4, "quality_of_limitations": 3, "questions_for_rebuttal": "-\tCan you please extend the literature section including other approaches such as latent force models?", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: **Post-Rebuttal Guidance**
Guidance: ACCEPT. The author rebuttal was very strong, with precise clarifications on what exactly the contributions are, additional comparisons to baselines and ablations. There was 1 strong accept and 1 weak accept. The reviewer (**FvMB**) who provided a weak reject has not acknowledged/responded to the rebuttal nor updated the score - I had posted my thoughts to reconsider their score in light of the rebuttal.

I would like to convey this message to the authors, which was brought up by the reviewer (**dC7P**): 
"The authors should also discuss and cite other recent works in the robotics community that have used PINN, for planning [1], and control [2,3].

[1] NTFields: Neural Time Fields for Physics-Informed Robot Motion Planning [2] Neural Optimal Control Using Learned System Dynamics [3] Value Iteration in Continuous Actions, States and Time [4] Physics-Informed Neural Networks for Continuum Robots: Towards Fast Approximation of Static Cosserat Rod Theory"

**Pre-Rebuttal Summary**

Strengths
- Elegant solution
- Strong quantitative evaluation, including on the real world

Weaknesses
- Limited technical novelty
- The presentation of the paper could be more clear
- Questions on generalization to more complex real data (not 3d printed shapes for example)

Dear authors, looks like the reviews are mixed and have several clarifying questions - please take a look.
Decision: Accept