Autoinverse: Uncertainty Aware Inversion of Neural Networks
Authors: Navid Ansari
Hans-peter Seidel
Nima Vahidi Ferdowsi
Vahid Babaei
Venue: NeurIPS 2022 Accept (2022)
Keywords: Neural networks
inverse problems
uncertainty
deep ensembles

Abstract: Neural networks are powerful surrogates for numerous forward processes.
The inversion of such surrogates is extremely valuable in science and engineering. The most important property of a successful neural inverse method is the performance of its solutions when deployed in the real world, i.e., on the native forward process (and not only the learned surrogate). We propose Autoinverse, a highly automated approach for inverting neural network surrogates. Our main insight is to seek inverse solutions in the vicinity of reliable data which have been sampled form the forward process and used for training the surrogate model. Autoinverse finds such solutions by taking into account the predictive uncertainty of the surrogate and minimizing it during the inversion. Apart from high accuracy, Autoinverse enforces the feasibility of solutions, comes with embedded regularization, and is initialization free. We verify our proposed method through addressing a set of real-world problems in control, fabrication, and design.

Autoinverse: Uncertainty Aware Inversion of Neural Networks
Navid Ansari
Hans-Peter Seidel
Max Planck Institute for Informatics
Max Planck Institute for Informatics
Saarbrücken, Germany
Saarbrücken, Germany
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hpseidel@mpiisb. mpg.....p..... de
Nima Vahidi Ferdowsi
Vahid Babaci
Max Planck Institute for Informatics Saarbrücken, Germany nvahidi Seiimmmpiiin mpg.de
Max Planck Institute for Informatics Saarbrücken, Germany
7babaei@mpi-int .mp.de
Abstract
Neural networks are powerful surrogates for numerous forward processes. The inversion of such surrogates is extremely valuable in science and engi- neering. The most important property of successful neural inverse method the performance of its solutions when deployed in the real world, i.e., on the native forward process (and not only the learned surrogate). We propose Autoinverse, a highly automated approach for inverting neural network surrogates. Our main insight is to seek inverse solutions in the vicinity of reliable data which have been sampled form the forward process and used for training the surrogate model. Autoinverse finds such solutions by taking into account the predictive uncertainty of the surrogate and minimizing it during the inversion. Apart from high accuracy, Autoinverse enforces the feasibility of solutions, comes with embedded regularization, and initial- ization free. We verify our proposed method through addressing real-world problems in control, fabrication, and design. Our code and data are available https://itlab..mpl http:/lla.mmiikkks.mmm. mpg. htt.am/i.......eeaaaaiiiia/aaaae de/nnansari/auuoonnvesss
of
Introduction
"... optimizingg for the wrong thing offering prayers to the bronze snake of data rather the larger force behind it." [6]
With the deep learning breakthrough during the last decade, data-driven modeling using neural network based surrogates is now a standard practice in science and engineering. These surrogates strive to imitate the behavior of a native forward process (NFP), such as physics simulation, which maps design into its performance While forward processes are essential for troubleshooting and analysis, oftentimes their ultimate application lies in their inversion, i.c., the reverse process of mapping performances into designs. Despite the recent progress inversion neural networks within multiple disciplines R5 16 13 37,, fundamental unaddressed question is still standing out. Inversion of surrogate model, fitted the data sampled from the NFP, ultimately different than the inversion of the NEP itself. The source of this gap could be explicit, such as the noise in
In the applications showcased in this paper cibrricatton-orientte design and robotics), the term design refers the space where the input to the forward process is characterized and parameterized and performance refers the parameterized space of desired properties. Commonly, hidden state or parameters, measurement or goal are used interchangeably with design and performance, respectively
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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measurements, or implicit, such as the poor sampling of the NFP. Although the obtained solutions from inverting the surrogate can be evaluated on the NFP, none of the current neural inversion methods offers tailored solution for addressing this important gap.
Our main insight in this work is expect and account for any potential mismatch between the data, and consequently the surrogate, on the one hand and the NFP on the other. Our proposed method, Autoinverse, realizes this vision by taking into account the predictive uncertainty of the surrogate and minimizing it during the inversion. Therefore, the inverted solutions avoid the unreliable regions within the training data.
We show that our Autoinverse strategy can augment existing neural inversion methods (both optimization- and trchitecture-basee approaches) with uncertainty compensation in simple and practical manner. Autoinverse closes the gap between the surrogate and the NFP not through attempting a perfect fit of the surrogate the NFP, an onerous task, but by finding inverse solutions in the vicinity of the reliable training data where the surrogate and the NFP are most similar. Neural inverse methods equipped with Autoinverse outperform their counterparts significantly on both standard data sampled from the NFP and imperfect data, e.g., those corrupted by noise. Apart from high accuracy, Autoinverse enforces the feasibility of solutions, comes with embedded regularization (freeing the inversion approaches from hand-crafted regularizations based on domain knowledge), and is initialization-free It achieves all these properties in a highly automated manner and only with light, intuitive tuning.
2 Related work
Neural Network Inversion We can divide neural network inversion approaches into two main categories. First, inverse architectures, where we compute a network architecture that takes a given performance and maps it into a (distribution of) design(s). Second, direct optimization [33], where we optimize for design such that it produces the desired performance. Although the simplest inverse architecture can be attempted by training a neural network in the reverse direction, it fails because of the one-to-many nature of the mapping. Tandem networks use an inverse architecture by employing pre-trained forward network in order to compute consistent loss. The tandem approach has been developed independently across different disciplines 23
39 dating back (at least) to Tominaga 34.. Many inverse architectures try to model the conditional posterior, p(x yy, using variational methods 24 [9] based on (conditional) variational auto-encoders [20). These networks condition the design on the target performance and yield distribution of solutions from which multiple samples could be drawn. Kruse et al. [21] show that the invertible neural networks (INNs), built upon normalizing flows [00,, give the highest accuracy in terms of both surrogate error and design posterior compared to a wide range of inverse architectures.
32 33
When using direct optimization methods, gradient- based optimizers can be readily used as the neural surrogate is differentiable. Ren et al. [30] use stochastic gradient descent via backpropagation with respect to the design variables, and present highly accurate and practical method. They benchmark their method, dubbed as neural adjoint (NA), against set of inverse architectures and obtain significantly more accurate solutions. Sun et al. (33 showed similar approach except using a quasi-Newton method for optimization. Ansari et al. [2] push forward in this direction by demonstrating that, for piecewise linear neural networks, e.g., those with ReLU activation, the direct optimization can be formulated as a mixed-integer linear program (MILP) and thus obtain globally optimal solutions. While optimization methods are very accurate, their main disadvantage is their performance as they can be orders of magnitude slower than inverse architectures.
Neural Networks and Predictive Uncertainty While neural networks are ubiquitous in almost all branches of natural sciences, their weakness at quantifying predictive uncertainty impedes their use in crucial applications. Using Bayesian formalism 4. Bayesian neural networks (BNNs) 28] 9 26). given the training data and a prior over network's 's parameters, compute the posterior distribution of the parameters. Having computed the posterior, the predictive uncertainty can be computed. The inference step in computing BNNs is known to be computationally hard [T7] This explains the popularity of simpler methods for estimating predictive uncertainty, such as Monte Carlo dropout [1]] and Deep Ensembles (22) Deep Ensembles strikes a good balance between simplicity and practicality on the one hand and predictive performance on the other hand (Section 3.3). One of the main advantages of the Deep Ensembles is its capability to predict alcatoric and epistemic uncertainty separately.

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|Pal (x*")) -y11
9/1 3xx
dc 0.1
oF + ((")))
Target performice y
Predicted performance
Design X
Ensemble
Inverse network
Ensemble Fule
Minimize
Design X
Forward Network
Forward Network
(a)
(b)
Figure 1: By using a deep ensemble predictive uncertainty estimator as the forward model we can make conventional inversion methods uncertainty aware. On the left
UANA and the
right UA-tandem architecture.
Aleatoric and epistemic uncertainty carry different information regarding the surrogate [8 and considering both of them improves the quality of the neural inversion (Section 4.3). Neural networks capable of predictive uncertainty are increasingly adopted in many different applications, such as
reinforcement learning and active learning [12]. As we shall see, neural inversion yet another domain that takes ad advantage of this trend.
Method
Autoinverse is an easy-to-implement technique for augmenting neural inverse methods with uncer- tainty awareness. Autoinverse achieves this goal by, first, training a surrogate capable of predictive uncertainty (25 Second, relying on this trained surrogate and using a novel inversion func- tion, Autoinverse finds accurate designs with minimal uncertainty. We apply Autoinverse on two inverse methods belonging the two main neural inversion categories, i.c., optimization- and architecture-base in Sections 3.1 and 3.2, respectively. As we will see in Section 3.3
we rely on established methods 22 train networks equipped with predictive uncertainty.
3.1 Uncertainty aware neural adjoint (UANA)
Given pretrained neural surrogate f(-), neural adjoint (NA) (30 is inverse method that uses the cost function CNA())) to push designs x to have performance f(x as close as possible to desired performance y":
NA = arg min [(xx - -x01
(I)
NA uses gradient descent iteratively reduce the cost function, scheme much like training neural networks but with the input as the optimization variable instead of network's weights and biases. Equation shows single NA iteration with the step size:
AL 0f
af
=
(2)
Ox
Autoinverse proposes to perform the inversion using pretrained BNN. We Deep Ensem- bles (22 made of neural networks capable of a prediction (x),, as well as its aleatoric Foa(xi and epistemic (xx uncertainties (Section 3.3 Aleatoric uncertainty increases the noise level in the training data increases. Epistemic uncertainty measures the uncertainty in the model. Autoinverse modifies NA such that obtain solutions that have performances close the target performance y* while resulting in small aleatoric and epistemic uncertainties:
CUANA == arg min [[F,(x)] "1, + F (x) + B FxE
We introduce a and 3 as hyperparameters to adjust the relative significance aleatoric and epistemic uncertaintics, respectively.

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Equation 4 shows how one iteration of UANA requires the back proppaapagn using the ensemble of all gradients M networks of Deep Ensembles:
OCUANA
O
(4)
0fm
Ox
where I'm represents of the networks in the ensemble. Figure ((a) depicts this collective procedure where each individual network in the ensemble votes for the direction where updating the design will lead to the maximal accuracy and minimal uncertainty.
3.2 Uncertainty aware tandem (UA-tandem)
Tandem is representative of architecture-basee methods in which we train an inverse network f-) in manner resembling the encoder-decoder architecture (B2 B4) Unlike the encoder-decoder approach, we start with training the forward model f(-). Then we freeze the trainable parameters of
f()) and train the inverse model f-)(-) in the position the encoder in order to decrease the cost function:
CJ CF == arg min ((f - y
(5)
f-1(-
Once f-)()) is trained we can simply query it to find designs with our desired performances:
(y))
(6)
The imcertainty-aware tandem (UA-tandem) follows the same procedure except that replaces f(-) with F (-). Additionally, it includes the uncertainties in the loss:
AUAT (f (y") arg min f-
- +a +a -0"
(7)
Figure [(b) depicts the architecture of UA-tanden
3.3 Predictive uncertainty using Deep Ensembles
Deep Ensembles comprises of an ensemble of M neural networks fm each capable of prediction Mmm (x) and its associated uncertainty om (x) in form of a Gaussian distribution N (pm (x), am (x)). The cost function for training
.E
each network in the ensemble the negative log likelihood
29
(o (x))
+ (y" - (m)))
CNLL
(8)
2
202 (x)
Intuitively, in case of the aleatoric uncertainty pm (x) fails to reliably predict y Hence, o (-) must increase to reduce the loss while the first term ensures o% (-) does not diverge to infinity.
The next step is ensemble the results of all the networks into single prediction and a single uncer- tainty. Deep Ensembles [22] models the ensemble as a single Gaussian distribution (IF (x)) approximating the mixture of M previously computed Gaussian distributions
Fx(x): = M Σ
Mm (x).
(9a)
F (x)
Σ (x) -F)
(9b)
M
The uncertainty of the ensemble can be decomposed into two input-dependent uncertainties, i.e., aleatoric F.A (x) and epistemic IF, (x) through slight modification to Equation 9b [18..

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Fo(x) = a Fx)(x)) + +B Fo)
(10a)
Faa(x): : M Σ (x),
(10b)
1
FxE(x) = M
(x( (x)).
(10c)
m
This enables us to control the behavior of the neural inversion by tuning the relative importance of these two uncertainties.
4 Evaluation
We evaluate the performance of Autoinverse through experimenting with the existing neural inverse methods and their uncertainty- counterparts. In the paper, we focus on NA and UANA while tandem and UA-tandem are evaluated mainly in the Appendix. We perform the evaluation on set of applications in robotics, and fabrication-oriente running the experiments 3 times to capture the variations. More details are provided in each case study and in Appendix (Section A)
computational design. We report the error by
Equations and have two hyperparameters (a and 3) that keep the balance between the MSE, the aleatoric uncertainty and the epistemic uncertainty. We observe that with a relatively larger epistemic weight (3) obtain better results. Exploiting this intuition, we tune these parameters for 3 different sets of values for {a, B/: {(0.1, 1}. {1. 10}, {10, 100)}. We then make two finer step depending on the outcome of the former evaluation and choose the best set of weights. We keep this budget of experiment runs for the rival methods as well. Typically, we use 10% of the target performance for tuning our inverse methods.
4.1 Experiments
Multi-joint robot is simple inverse kinematics problem which is being used standard test for neural inverse problems B0] In this problem, the design is the ID position of the base of the multi-joint robot along with the ID rotation of its three joints. The inverse problem concerns finding a combination of positions and angles for the base and the joints such that the tip the arm lands the target position, i.e., the performance. We follow R0 for setting up this experiment and its corresponding analytical equation used as the NFP. The training data consists of 10,000 pairs of samples generated by randomly sampling the NFP.
Spectral printer Spectral printing enables digital fabrication of the object's 's appearance faithfully ()). Unlike reproduction of the color (e.g., RGB), reproducing the spectrum ensures that the original and the duplicate remain visually similar independent of the color of the light source. Spectral printing has various important applications specially in fine art reproduction using both 2D and 3D printing (2 i 32 Deep neural networks are becoming the main computational tool for modeling the spectral printing process specially when dealing with large number of inks. The final objective to find the correct ink densities at each pixel that can best estimate the target 31D spectrum.
In this experiment, we create the NFP by simulating a printer using an ensemble of 20 neural networks. The design space comprises the ink densities and our spectral printer NFP predicts the spectrum of the resulting color. We use real, measured data from an 8-channel printer with 8 EPSON inks including Cyan (C), Magenta (M), Yellow (Y), Black (K), Light black (LK), Light light black (LLK), Light Cyan (LC), and Light Magenta (LM). The light inks, added to the standard CMYK to improve the print quality, introduce significant muuli-modallity All networks in the ensemble NFP are trained on 40,000 printed patches [I] consisting of different ink-density combinations and their corresponding spectra. The ensembling is intended for an accurate NFP and is independent of our use of Deep Ensembles for computing the uncertainty. The visual nature of spectral printer makes attractive for analyzing different methods. We release this NFP to the public to add another example to the neural inversion testbed.
5

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Table 1: The NFP and surrogate enrors (mean = STD) of different neural inverse methods on multi-joint robot for 1000 target locations.
Error
NA
UANA
tandem
UA-tandem
MINI
INN
NFP
(3.24 +: 0.51)
(3.21 + 1.48)
(4.42 +: 1.56)
(4.04 - 0.38)
1.6
(9.48 - 0.021)
x x10-1
x10-6
x10-3
x10-
X x10-3 0
x10-3
Surrogate
(1.99 + 0.05)
(9.13 + 6.08)
(8.58 + 3.00)
(7.10 0.64)
(2.04 = 0.017)
x10-
x10-7
x10 6
x10
x10
Soft robot are made of soft, flexible materials. This unique property has made them suitable candidates for interaction with humans in applications like minimally invasive surgery and advanced prosthetics [7]. Unlike multi-j
robot with limited number of rotating joints, each segment of the soft robot a potential actuator that through their contraction and expansion can determine the robot's is final shape. This inverse kinematics problem is typically solved through partial differential (38) ). order to accelerate the solve time of this inverse kinematics problem, Sun et al. 33 proposed a neural surrogate modeling of the problem and its inversion via tandem. The design space in this problem consists of the contraction or expansion of 40 controllable soft edge segments. The superposition of all the actuations determines the final deformation position of the soft robot via the position of its 206 vertices.
equations
as our NFP. We model the relationship between the actuations and the final shape of the soft robot with a neural-network surrogate. Our goal solve the neural inversion to find a suitable set of actuations (design) that brings the tip of the soft robot to the target position (performance). The training data consists of 50,000 samples queried by random The designs are then
We use an FEM- simulation (38 15
sampling the actuation with an expansion ratios between -0.2 and 0.2 33 evaluated on the TEM-based NFP to calculate their deformations.
4.2 Quantitative comparison of different neural inversion methods
We evaluate the accuracy of a set of neural inversion methods on multi-joing robot in terms of both the NFP and the surrogate errors. The surrogate error is the difference between the 're-prediction' of the obtained inverse solution by the surrogate neural network and the target performance. The NFP error is the difference between the target and the performance of the generated design evaluated on the NFP. In addition to the core methods described so far, we evaluate mixed-integer neural inversion (MINI) and the invertible neural network (INN) (3)
Table summarizes the inversion results on 1000 randomly sampled target locations for the multi-joint robotic arm. We keep our evaluation fair by setting the capacity of the neural surrogates comparable wherever possible. For instance, all methods except MINI have around 3 million parameters (see Appendix, Table 3 for more details). We also used equal computational resources for the tuning. Methods with hyperparameters, like UANA and UA UA-tandem, are tuned in 5 stages by hand. Alterna- tively, the methods without hyperparameters (NA, tandem) are given 5x extra budget for inversion: We run NA and tandem 5x and choose the model that generates the best NFP error. We repeat this process 3x and report the standard error. MILP and INN are fundamentally different methods. MILP finds the global optimum and thus does not need tuning. INN has a latent space which we can sample to generate diverse designs. We sample INN's latent space 1024 times for all 1000 targets, evaluate them on the NFP, and report the best NFP error. We repeat this process three times to generate the standard error.
Table shows the outstanding accuracy of the inverse methods that adopt Autoinverse, i.e., UANA and UA- tandan,,, in terms of the NFP error. It also demonstrate how even a perfect surrogate error (c.g., MINI) does not guarantee accurate solutions when tested on the NFP. A second look at Table [ reveals further interesting insights. Although NA obtains notably lower surrogate error than its uncertainty-await counterpart (UANA), it performs significantly worse in terms of the NFP error. The main reason for this trend is that, when optimizing the surrogate, UANA is not only concerned
with finding accurate designs leading to a small accuracy gap between the target and candidate performances, i.e., the surrogate error, but also with those designs featuring low uncertainty (through the uncertainty term in Equation T0). Therefore, UANA achieves high accuracy in terms of the essential NFP error at the cost of worsening the inconsequential surrogate error. Furthermore, UA-tanden for example, achieves better performance than NA. This is remarkable result for cesttecture-based

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Table 2: The distribution of ink densities (2 0.4) after the inversion of spectral printer using UANA. Once we insert noise into LC channel or sample it sparsely, Autoinverse detects and avoids it. STD has been rounded the nearest integer.
Model
dataset
NFP
Y
LC
LM
LK
LLK
Standard
(6.30 + 0.081) 10-)
67
63
437
DAHA
Spunse
(5.64 1: 0.017)
316 1:1
60
50
2::00
826 1:5
25
323 11
Noisy
[6.13 4: 0.026)
10
263
67 4
34
1::0
0
276 3:3
29 0
378 378
Standard
0.001)
MA
Spanse
[1.34 + 0.007)
x 10"1
905 +-6
606 +
1515 +-8
137 +-6
1604 11
2130 + 17
204 +
23112
Noisy
(1:47 0.002) 10
1192 2: 14
988
1128 4:
:
1029 23
9:1
283
1742
Edge 16
Edge
Edge
Edge
Edge 21
Edge 25
Edge 31
Edge 39
Noisy
Sparse data
Figure 2:
Distribution of actuation values for soft robot. Edge 16 is corrupted by noise (top) and sampled sparsely (bottom) in positive range. While other randomly chosen edges feature both negative and positive actuations, UANA produces solutions without (or with less) positive actuations for Edge 16.
method given is significantly faster than the optimization-based NA. We report extensively the time performance and the details of the training for this and the next experiments in the Appendix.
4.3 Neural inversion in the presence of imperfect data
One of the main advantages Autoinverse appears in scenarios where the training data suffers from noise, e.g., measurement noise or poor sampling, e.g., sparsity in some regions. We evaluate the performance of Autoinverse on imperfect training data using NA and UANA (tandem and UA-tandem evaluated under the same configuration in the Appendix Section C).
Locally sparse data We sample both soft robot and spectral printer NFPs (Section 4.1 way that the data does contain any samples from one of the inputs specific interval. For spectral printer, we would like to find ink densities to reproduce the spectra of the colors in the painting in Figure (made of 3568 distinct color spectra). We sample the printer channels (no ink), 0.05, 0.1, 0.5, and (full ink) densities to form the standard training data (with no known uncertainty). We create a partially sparse dataset similar the standard one except for the Light Cyan (L.C) channel for which we only have samples at O. 0.05 and 0.1. Table 2 shows that while for the standard dataset UANA finds inverse solutions that include the LC channel frequently (437 times), for the sparse dataset it avoids this channel completely and compensate for it using the Cyan channel. UANA is able to avoid this channel as the epistemic uncertainty increases in sparse regions dataset (see Appendix Section C.4 more details).
In soft robot sample the 16th (among 40) controllable edge only in the negative range (con- traction only). We then use the trained network to invert 1000 test samples. Figure 2 shows the distribution of each edge for 1000 inversion tasks. We have plotted the distribution for the 16th edge as well as for 7 other randomly chosen edges. As evident from Figure bottom row, UANA is highly reluctant to choose designs with positive actuations for this edge.
Locally noisy dataset With the same problem configuration as before, we would like to test the robustness of Autoinverse on a dataset locally corrupted with noise. We start with standard dataset and inject Gaussian noise (0, 0.1) to the spectrum of the samples with more than 0.4 LC density. 2 shows how after introducing noise to the LCC, the network avoids that channel and compensates it by using more Cyan instead. In soft robot corrupt the final shape of the soft

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Ground tuuhh
(MSE 0.1400)
design
UANA NA with boundary loss
NA with boundary loss (MSE 0.1148)
UANA (MSE 0. (,0087)
XXX
Figure 3: In painting reproduction, UANA outperforms NA with and without a boundary loss term. The distribution of the inverse solutions shown on right.
robot with Gaussian noise (0, 0.5) for all the shapes where the 16th edge has positive actuation. As we can see in Figure 2 (top row), the inversion has completely avoided positive actuations for this edge.
4.4 Autoinverse brings AutoML to neural inversion
Autoinverse incorporates feasibility. Deep Ensembles produces high epistemic uncertainty outside the distribution of the training data. This includes the regions where the NFP is not defined and thus not sampled. For example, ink densities outside [0, are not printable. If such cases arise during the inversion, they are clipped to j0, 1] [.] In such regions, networks in the ensemble do not agree and thus the epistemic uncertainty increases. Autoinverse automatically avoids these unfeasible regions. In order to simulate the result of the inversion, after clipping the out-of-range densities we feed them into the NFP. As evident from FigureBJleft, inversion via NA results a poor reproduction of the original painting. UANA, on the other hand, achieves spectacular reproduction accuracy. This is explained by the plot in Figure right, showing the distribution ink densities obtained from both methods.
Autoinverse has built-in regularization. We incorporate regularization into inversion methods in order to obtain solutions that, among other purposes, agree with the observations and follow a certain statistical distribution. Oftentimes, regularization comes with unexpected side effects. Here we show that Autoinverse follows the distribution of the training data naturally by taking into account the epistemic uncertainty. We validate this point using both spectral printer and soft robot experiments. We show that Autoinvers without any explicit regularization performs better or on par with its counterpart inversion methods equipped with regularization.
case-specific, requires human knowledge, and
In the spectral printer experiment, we evaluate the effect of the boundary loss, originally proposed as a generic regularization for NA to limit the designs within a box constraint (see (30) and Appendix Section D) The boundary loss added to Equation weighted using a hyperparameter. We tune this parameter with the same tuning budget we allocate for tuning the uncertainty weights (5 set of inversions on evaluation data). In Figure we observe that although NA with boundary loss improves the distribution of ink densities within the valid region (00, 1). it still trails the regularization- free UANA significantly.
Regularizing the soft robot is less intuitive as the superposition of all actuations determines the final shape and whether is physically plausible. In 33). the regularization is smoothness term that keeps actuation values near each other (see Appendix Section D). Figure 4(a) 1) demonstrates how NA fails without regularization to control the robot with reasonable deformation. Once the regularization is added to NA, the designs become physically meaningful (Figure ()). Figure
4(a) 3) shows how UANA performs comparably without any regularization.
8

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Correct initialization (NA, UANA)
Frequency
Wrong initialization (NA, UANA)
-0.5
0.5
3
2
Expansion Contraction (x)
(a)
(b)
Figure 4: On the left we can see the distribution of actuations calculated by UANA and with two different initialization: one initialization near the training data distribution and one far from it. On the right we can see a range of randomly chosen soft robot shapes calculated by different methods with different regularization and initialization.
Autoinverse is initialization-free Sensitivity to the initialization is widely known issue in non- convex optimizations. Despite equipping NA with the smoothness term soft robot, Figure (4(b demonstrates how incorrect initialization can result in solutions with seemingly good surrogate and regularization loss (see Appendix Section D but in the infeasible region the design space (Figure (4). UANA with the same incorrect initialization and without any regularization, leads to soft robot designs that reach the target location accurately and produce plausible deformations (Figure 4(a) ()). For UANA, the solver starts with reducing the main contributor to the optimization objective, lee, the epistemic uncertainty. Once a region with a reasonably small uncertainty is reached, the accuracy term starts to take effect and desirable solution within the valid range of the design space found.
4.5 Ablation studies
Separating the ensembling effect Although in both and UANA we use surrogates with similar capacity (network size), could argue that the higher performance of UANA comes from the ensemble architecture of its surrogate. We detach the impact of ensembling and uncertainty awareness on the inversion performance by implementing NA ensemble where, instead of single, large forward neural network, it uses an ensemble of networks. The inversion procedure identical to NA (see Appendix Section E and Figure 5). We employ NA ensemble in multi-joint robot with the same configurations as Section 4.2 The surrogate and NFP error for NA ensemble are (3.30 = 0.59) x 10 and (1.17 + 0.32) 10 respectively. Comparing these values with those in Table I. NA ensemble shows a slight improvement over NA but is significantly outperformed by UANA.
Diversity of activation functions in Deep Ensembles Deep Ensembles uses similar set of networks with identical activation functions [22]. In practice, we observe that a diverse set of activation layers leads better performance of Autoinverse. Different activation functions generate different behaviours, show higher disagreement where the training data is umder-represented and, thus, result in accurate estimation of the landscape of epistemic uncertainty [36]. For all experiments in this work, we use a diverse range of activation functions (see Appendix Section A). As ablation, we UANA on spectral printer using sparse data (same configuration as Section (.3) but with ReLU as the only activation layer. In contrast to Table 2 UANA (with ReLUU only activation) does not completely avoid the sparse domain and delivers 19 + solutions that contain LC with densities 2 0.4.
5
Discussion
The Autoinverse cost function is multi-objective. Instead of finding a single solution through weighted combination of these objectives (what we have seen so far), we can capture the trade-off between the accuracy and uncertainty through computing the Pareto front (see Appendix Section

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for more details). Autoinverse is an inversion strategy that could be applied to various inverse methods, especially those that concemn imitation of original process (dubbed NFP in our work). We look forward to see Autoinverse adopted for more inverse architectures beyond tandem, and more direct optimizations beyond first-order NA.
Predicting the uncertainty using Deep Ensembles [2 is not the most efficient solution. This is because multiple forward networks are trained in two stages (once for the mean, and once jointly for the mean and variance). It is highly interesting to integrate more efficient uncertainty estimation methods, such as Monte Carlo dropout
[8], into Autoinverse.
Acknowledgments and Disclosure of Funding
We are grateful to Xingyuan Sun and Szymon Rusinkiewicz who provided the repository of the soft robot simulation [3], Sebastian Cucerca for the revision of the figures, and anonymous reviewers for valuable feedback. Special thanks to Jalil Maani and Soraya Iranpak for their kind support.
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12


Reviews:
[{"review_id": "r7IQUI2beaw", "reviewer": "Reviewer_ijXf", "summary": "(Updated score post rebuttal)\n\nThe paper proposes a new method AutoInverse in the field of inverting neural networks that incorporates the uncertainty in the modelling of the NFP (natural forward process) though a neural network surrogate. By adding the epistemic uncertainty of the surrogate function (due to mismodelling) and aleatoric uncertainty in the training data (noise in the data), the new method favours points that have low uncertainty, and therefore are a better match to the NFP. Incorporating uncertainty in the inversion loss also results in desirable behaviour, such as 1) enforcing the feasibility in solutions, as points with low feasibility should have high uncertainty, 2) not requiring regularisation techniques, and 3) not requiring careful initialisation. ", "questions": "1. I feel like the authors need to incoporate a few more papers in their literature survey regarding the modelling of aleatoric and epistemic uncertainty using a heteroskedastic regression loss. The decomposition shown in the paper has been shown before, in [1]. There has been work also showing that using this formulation can have pathological behaviour [2]. \n2. I found this paper lacking an adequate explanation of the tradeoff between aleatoric and epistemic uncertainty terms. The authors mention that \"We observe that with a larger epistemic uncertainty we obtain better results\", but I think it's important to have a better ablation here. An easy way of \"improving\" epistemic uncertainty estimates is to use more deep ensembles, and see how increasing the number of ensembles affects the choice of $\\alpha$ and $\\beta$ parameters. Similarly, by adding more noisy data, in the \"Locally noisy data set\" ablation, one can measure the importance of the aleatoric term. I think having these two sweeps in the ablation, along with $\\alpha$ and $\\beta$, will really help understand the differences between the two terms.\n3. Given that the main contribution of this paper is a way of incorporating uncertainty estimates into the inversion cost function, I think in order for the paper to be a good candidate for acceptance, there needs to be a better ablation for different uncertainty estimation methods. I've mentioned more of this in the Significance section.\n\nMinor nits:\n- Line 75: \"neural\"\n- Line 112: \"push\"\n- Line 313: \"similar\"", "limitations": "I believe the authors should address the additional compute requirements needed for their method. Deep ensembles are the least efficient way of obtaining uncertainty estimates, and can result in significantly more GPU hours for training and inference. These should be addressed.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "1. Originality\n\nThe Autoinverse technique has two components: 1) Replacing the surrogate neural network function with a neural network function that is capable of predictive uncertainty, and 2) modifying the loss function used to find accurate designs given a pretrained surrogate function with frozen parameters.\n\nThe first component of this technique is a well-established baseline in literature, and deep ensembles have been very well studied in their ability to accurately model predictive uncertainty in a bunch of different domains and tasks. In fact, the decomposition of the uncertainty learnt using deep ensembles (or other methods that model heteroskedastic regression) into components that model aleatoric and epistemic uncertainty are also well-studied [1]. However, the authors do not cite [1], or discuss any potential pitfalls of modeling noise in such a way (see [2] as an example.)\n\nThe second component is the novel contribution for this paper, in my uncerstanding. By using a neural network that is modeling both heteroskedastic aleatoric uncertainty and epistemic uncertainty, this method can use these estimates as auxilliary terms in the inverse cost function, to directly influence the training of the designs. By adding epistemic uncertainty to the inverse cost function, points that have high epistemic uncertainty are downweighted (in a sense), and these are usually points where the surrogate is not performing well or is performing pathologically. By adding aleatoric uncertainty to the cost function, points with high data noise are also downweighted. This simple addition of two terms to the cost function results in quite impressive empirical performance.\n\n2. Quality\n\nIn my opinion, the paper is well-written, has good motivations for the problem it is tackling, and has strong ablations that answered quite a few of my initial doubts/hesitations about the empirical strengths of the technique. \n\nI have some concerns regarding the thoroughness of uncertainty quantification methods studied, and a careful analysis of the strengths and weaknesses of different methods. For example, Deep Ensembles are known to be state-of-the-art in uncertainty quantification, but have much higher compute time and storage requirements than other methods. The authors mention that the longer training time for ensembles is not a concern, as they are trained parallely. I disagree with this point, as I feel like acknowledging the GPU hours required to train a method are quite important, especially with big compute quite easily getting out of hand to obtain SOTA. Even though there are certain groups with the resources to trivially parallelise ensembles, if Autoinverse is to be established as a strong contender, the added compute time, memory footprint, and inference time needs to be acknowledged. \n\nAnother factor here is maybe exploring a few other uncertainty quantification techniques as an ablation in the paper (MC Dropout, more efficient ensembles etc.). I feel like this comparison is important to understand the contribution of adding an aleatoric and epistemic uncertainty term to the cost function, vs the quality of these uncertainty estimates w.r.t the compute required. It might be possible that deep ensembles are overkill for the inversion problem, and a lot of compute is used to obtain uncertainty estimates that might be approximated more cheaply. As the authors mention, it is a very interesting idea to capture the Pareto front between accuracy and uncertainty in an inversion task.\n\n3. Clarity\n\nThe paper is quite easy to follow, and the experiments chosen make logical sense. I would argue that it is more important to describe the inversion technique and cost function before talking about uncertainty quantification, as it is maybe more prudent to motivate the exact problem that the authors wish to tackle. In the current ordering, it might seem that the authors use a well-known methodology of uncertainty quantification, and find a usecase that works well. Though this is obviously not true, a reordering of the sections might help mitigate this.\n\n4. Significance\n\nUncertainty quantification in deep neural network applications is of fundamental importance. In order for DNN methods to be applicable in safety-critical domains, it is very important that we measure their performance, calibration and out-of-distribution robustness. To my knowledge, there has not been prior work that adequately addresses the surrogate-NFP mismatch when formulating the cost function for inversion. This paper's application is quite significant, and I believe that the empirical results show that modelling uncertainty in a principled way can result in multiple benefits. However, I feel like this paper is lacking in terms of adequately populating the Pareto front between uncertainty and accuracy, by proposing a single data point (ensembles). While the current proposed method cannot be argued with in terms of empirical performance, I feel like this paper might be a good place to populate more points here, to establish a better understanding for the community, on how the quality of uncertainty estimates affect the performance of inversion.\n\n[1] Kendall, Alex, and Yarin Gal. \"What uncertainties do we need in bayesian deep learning for computer vision?.\" _Advances in neural information processing systems_ 30 (2017).\n\n[2] Seitzer, Maximilian, et al. \"On the pitfalls of heteroscedastic uncertainty estimation with probabilistic neural networks.\" _arXiv preprint arXiv:2203.09168_ (2022).", "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": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "hHWznU85TRI", "reviewer": "Reviewer_uj48", "summary": "The paper discusses an approach for inverting surrogate functions, based on neural networks, in a face of (epistemic and aleatoric) uncertainty.", "questions": "1) Section 2 (Related work) would be expected to place  this work within the current state-of-the-art, however, it does not look like it fully does: how do the state-of-the-art works address the problem of estimation of uncertainty, what prior methods address this problem and what is the closest work which the authors build or improve upon? It is said that ‘In this method, a specialized architecture built upon normalizing flows [9] is trained   simultaneously in both forward and inverse directions leading to a bijective mapping between design and performance spaces. “ but it is not followed up in the following sections.  Furthermore, the similarity between the proposed ensemble architecture and normalising flows is unclear to the reviewer. (Also, on this sentence, the reviewer has probably misunderstood how exactly could the proposed modification of normalising flows be trained in both directions, if the original normalising flows [Resende &Mohamed, 2015] are invertible-by-design?)\n2) The reviewer struggles to follow section 3 (Method): what is the precise model architecture, for example? (while there is the code so that the model is reproducible, it is still important to discuss and justify main aspects of implementation) While Appendix gives implementation details (Section 1 of the Appendix), it is unclear what the relation to neural adjoint method is and how does the proposed model build upon it; furthermore, it would be good if the paper references the descriptions in the appendix. \n3) In Eq. 4-5, it is important to clarify, why the optimisation happens with respect to the inputs X, and where and how are the parameters of the model optimised.\n4) “Following [27], we take a 3-step procedure for implementing deep ensemble predictive uncertainty. “ What would the three steps be?\n5) On the experiments side, it is important to describe the experimental tasks in more detail to make it reproducible (which are not Ava; the reviewer thinks that the paper should describe what exact hyperparameters and settings correspond to the experimental evaluation.\n6) It is important to identify (and compare with) the state-of-the-art methods; if it is not possible for whatever reason, there should be a convincing argument why such comparison would not be valid. I accept that Table 1-2 present evaluation against Neural Adjoint method, but just two comparisons on two tasks do not indicate the advantages (and also importantly, restrictions) of the proposed method.\n7) (Not as critical as the previous points but still) The concluding remarks state: ''We look forward to see Autoinverse adopted for more inverse architectures, beyond tandem and more optimizations (beyond first-order NA). The Autoinverse cost function is multi-objective. It is highly interesting to capture the trade-off between accuracy and uncertainty using the Pareto front. '' It does not give much summary of the main conclusions of the paper, and also the future work suggestions are difficult to follow (more inverse architectures/optimisations: in which sense? Pareto front: would be interesting to see it, could be actually quite nice to see them as an additional experiment, e.g. in the appendix)\n", "limitations": "There is a need to discuss the limitations of the work more; one way to address it would be to use standard benchmarks such as the ones proposed in the standard invertible neural networks literature (e.g. MNIST and CIFAR experiments from Rezende and Mohamed, 2015, or experimental section of Grathwohl et al, 2019); otherwise, it would be useful to state why such experiments are not like-for-like or not possible.\n\n* Rezende D, Mohamed S. Variational Inference with Normalizing Flows. In: International Conference on Machine Learning, 2015 (pp. 1530-1538). PMLR.\n* Will Grathwohl*, Ricky T. Q. Chen*, Jesse Bettencourt, Ilya Sutskever, David Duvenaud. \"FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models.\" International Conference on Learning Representations (2019).", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "\nStrengths :\n\n- The problem statement: quantification of uncertainty for invertible processes is an important problem\n- The idea of modelling uncertainty as outlined in Section 3.1 is a reasonable one\n- The authors plan to release the spectral printer dataset; the new dataset would be beneficial for the research community\n\n\nWeaknesses:\n- The writing looks not entirely clear (see the question section)\n- Novelty could also be emphasised as it is unclear where the proposed method stand amongst the state-of-the-art models such as variational autoencoders and normalising flows (see q1 in the questions section)\n\nI put the score as 'reject' for now, but looking forward to the rebuttal and hope the authors could address the questions appropriately.\n\n***\n\nThe score is updated as the concerns have been thoroughly addressed during the rebuttal (see the discussion below).", "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": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "8zk6kfKP2ro", "reviewer": "Reviewer_xeN4", "summary": "The paper presents an approach to solving inverse problems, i.e. find parameters $\\mathbf x$ that explain/produce given observations $y$ when passed through the forward model $f(\\mathbf x)$. The approach explicitly models and accounts for the uncertainty of the forward process by learning a surrogate of it using a deep ensemble. The deep ensemble models the forward process as $y \\sim N(\\mathbb F_\\mu(\\mathbf x), \\mathbb F_\\sigma(\\mathbf x)$, with $\\mathbb F_\\sigma(\\mathbf x)$ explicitly modelling the uncertainty of the forward process. The   $\\mathbb F_\\sigma(\\mathbf x)$ is then plugged within two inversion methods, the neural adjoint and a tandem architecture, producing their uncertainty-aware variants. As a result when looking for the inversion solution the method will avoid $\\mathbf x$ which come with a high uncertainty with the $y$ that they produce. The two methods are evaluated in three benchmarks and demonstrate good inversion performance. ", "questions": "In relation to the single-point vs the complete posterior distribution it would be useful to discuss how the single point solution fits within the complete posterior distribution, can we make any other statement on where the point estimate x relates with the complete p(x|y) other than that it is an $x$ for which $p(y|x)$ has the least uncertainty?\n\nIn the evaluation measures I am not sure I get correctly what the surrogate error is and what the NFP error is. Is the surrogate error $|| f(f^{-1}(y)) - y||$ or is it $||f(\\mathbf x) - y||$, in other words does it measure the gap between the prediction of the forward surrogate in the inverse solution, or the gap on paired training instances (x,y). In what concerns the NFP error, is this the $|| NFP(f^{-1}(y)) - y ||$, i.e. the error we get when we pass through the real forward process at the inverse solution? \n\nWith respect to the INN wouldn't a more appropriate way to get the inverse solution be the MAP estimated of the p(x|y) instead of getting empirical samples from it?", "limitations": "The authors adequately addressed the limitations and potential negative societal impact of their work.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper is rather well written. The main idea of the paper is that when looking for inverse solutions, $\\mathbf x$, that could produce a given observation $y$ one should avoid $\\mathbf x$ that come with high uncertainty for their predicted $y$, as quantified by the deep ensemble and the estimated $\\mathbb F_\\sigma(\\mathbf x)$ parameter. In fact $\\mathbb F_\\sigma(\\mathbf x)$ can be decomposed in two terms one measuring aleatoric uncertainty and the other the epistemic uncertainty. The latter is the extend to which the components of the ensemble differ with respect to the average prediction, i.e. the $\\mathbb F_\\mu(\\mathbf x)$. A high epistemic uncertainty can be evidence of the $\\mathbf x$ strays away from the training distribution and thus the base models produce very different predictions. Incorporating the term in the loss functions keeps the search for an inverse solution away from such high uncertainty regions in which the learned surrogate is probably not a very accurate proxy of the true forward process. \n\nIf I understood correctly the paper only provides a point estimate, i.e. a single $x$ given a desired $y$, and along the above discussion the one with the least uncertainty. To my understanding in solving inverse problems one is also strongly interested in accessing the full posterior of $\\mathbb x$, i.e. $p(\\mathbb x|y)$, something that is delivered by generative methods such as the invertible neural networks which the authors cite and compare as one of the baselines. There is no discussion, unless I missed it, on the quite point estimate vs posterior distribution approach; this is a weakness of the paper, and one could also say a weakness of the method.  \n\n", "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": "", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept