Inductive Domain Transfer In Misspecified Simulation-Based Inference
Authors: Ortal Senouf
Antoine Wehenkel
Cédric Vincent-Cuaz
Emmanuel Abbe
Pascal Frossard
Venue: NeurIPS 2025 poster ()
Keywords: AI4Science
SBI
Domain transfer
sim2real
Primary area: machine_learning_for_sciences
TL;DR: We propose a semi-supervised domain transfer approach for bridging the sim2real gap in simulation-based inference, leveraging mini-batch optimal transport for scalable, inductive posterior approximation.

Abstract: Simulation-based inference (SBI) of latent parameters in physical systems is often hindered by model misspecification--the mismatch between simulated and real-world observations caused by inherent modeling simplifications. RoPE, a recent SBI approach, addresses this challenge through a two-stage domain transfer process that combines semi-supervised calibration with optimal transport (OT)-based distribution alignment. However, RoPE operates in a fully transductive setting, requiring access to a batch of test samples at inference time, which limits scalability and generalization.  We propose a fully inductive and amortized SBI framework that integrates calibration and distributional alignment into a single, end-to-end trainable model. Our method leverages mini-batch OT with a closed-form coupling to align real and simulated observations that correspond to the same latent parameters, using both paired calibration data and unpaired samples. A conditional normalizing flow is then trained to approximate the OT-induced posterior, enabling efficient inference without simulation access at test time. 
Across a range of synthetic and real-world benchmarks--including complex medical biomarker estimation--our approach matches or exceeds the performance of RoPE, while offering improved scalability and applicability in challenging, misspecified environments.

Inductive Domain Transfer In Misspecified Simulation-Basen Inference
Ortal Senouf
Antoine Wehenkel Apple Zürich, Switzerland
EPFL
Lausanne, Switzerland
Cédric Vincent-Cuaz
Emmanuel Abbé
Pascal Frossard
EPFL
EPFL, Apple
EPFL
Lausanne, Switzerland
Lausanne, Switzerland
Lausanne, Switzerland
Abstract
Simulation-barn inference (SBI) of latent parameters in physical systems of- ten hindered by model misspecification-the mismatch between simulated and real-world observations caused by inherent modeling simplifications. RoPE, a recent SBI approach, addresses this challenge through two-stage domain transfer process that combines semi-supervised calibration with optimal transport (OT)- fully transductive
based distribution alignment. However, RoPE operates setting, requiring access to a batch of test samples at inference time, which limits scalability and generalization. We propose fully inductive and amortized SBI framework that integrates calibration and distributional alignment into a single, end-to-end trainable model called FRISBI. Our method leverages mini-batch OT with a closed-form coupling to align real and simulated observations that corre- spond the same latent parameters, using both paired calibration data and unpaired samples. A conditional normalizing flow is then trained to approximate the OT- induced posterior, enabling efficient inference without simulation access at time. Across a range of synthetic and real-world benchmarks-inclue medical biomarker estimation-our approach matches or exceeds the performance of RoPF, while offering improved scalability and applicability in challenging, misspecified environments.
complex
Introduction
Inference of latent variables that describe important properties of physical systems is fundamental problem in many domains, including environmental [1(2), mechanical 3 [ 5) and physiological [6]
7 systems. Traditionally, this problem has been approached by formulating a mathematical model that relates the observations æ to the latent parameters of interest 0, and solving the corresponding inverse problem to infer 0 from æ [0].
Modern machine learning (ML) has achieved remarkable success complex tasks, sparking interest in its application inferring latent parameters from observations. However, standard supervised learning is often impractical in this context, as ground truth parameter data is typically expensive infeasible obtain, such in medical applications where direct measurement may require invasive procedures. To address this, two prominent approaches have emerged: simulation-base inference (SBI) (I] and hybrid learning [12] [3 [4]. SBI trains ML models simulated data to directly estimate parameters while capturing uncertainty through posterior estimation, becoming
"Corresponding Author, ortal. www.seeeeol...comm
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Joint OT supervised learning
x, E 190 catib
(2) Posterior mixture amortisation
Supervised
8 (
4
Σ
8$
B
(x)
(X,)
x, E
D wxpaired:
D catib
Model be trained
Trainable embeddiing x
Pre-tmined emberdding of Trained, domain-transfered embedding xf x,
Figure 1: FRISBI Overview. Similar to RoPE (22). we assume a trained neural statistics encoder
(NSE), hwr, that maps simulation data æ to embeddings h (æx.). and a neural posterior estimator (NPE), qa*, which estimates simulated posterior distributions. FRISBI performs: (1) Joint optimal transport (OT) and supervised learning. Both paired and unpaired samples contribute to the OT plan (dashed lines), weighted by aj. Supervised samples from Deatsh (solid lines) anchor the OT matching. The -oleeveervttoons encoder 96 is fine-tuned to optimize representations for both supervised learning and OT oT-based domain transfer. (2) A conditional density estimator, qç. approximates the posterior arising from the OT-bbased mixture of posteriors.
cornerstone in scientific domains 16 16 18 19,, though it can suffer from sensitivity model misspecification [20) (21) In hybrid learning, the simulator is integrated with ML components to obtain a more accurate model of the system. While this approach provides a useful inductive bias, often requires simulators that are differentiable and computationally feasible, limiting its applicability realistic and complex systems.
recent work introduces RoPE [22]. an SBI approach designed to address model misspecifications through a two-stage, semi-superriseer domain transfer strategy. The first stage focuses pointwise calibration using a small set of labeled real observations- i.c., observations for which the correspond- ing parameters are known, whereas the second stage aligns distributions using optimal transport (OT). While effective, this approach inherently transductive, it requires a batch of test samples for inference, limiting its applicability in scenarios that require inductive inference. In many world settings, access to a batch of test-time observation is unrealistic, and the inferred posterior for given single test input can vary depending on the batch it embedded in-undermining stability and reproducibility. Additionally, the strict separation between pointwise and distribution-win alignment may prevent full exploitation of their complementary strengths.
In this work, we propose a new framework, illustrated in Fig. [ that builds upon elements of RoPE by amortizing the optimal transport (OT) step through mini-batch unbalanced OT approach 23 24
Similarly to RoPE, our method assumes access to limited set of ground-truth pairs of observations and parameters O. It features closed-form solution for the transport plan and offers two key advantages:
Inductive Joint Training of Alignment Steps: Enables end-to-end training of both point- wise and distribution- wise alignment, better leveraging their complementary strengths.
Amortised Posterior Estimation: Provides scalable, inductive solution for OT-based posterior estimation, climinating the need to repeatedly access simualtions during inference.
We show that our method, while fully inductive and applicable to individual test samples, achieves competitive- and often poperior--prformannn compared to the transductive RoPE baseline across range of benchmarks. This includes both synthetic and real-world datasets, with strong results in terms of accuracy and calibration, even in challenging settings such as complex biomarker estimation.
Background
2.1 Simulation- Inference and Neural Posterior Estimation
We consider a simulator S 0 - that, given parameters p(()), produces simulated data x, S(0), whose likelihood p(z, is intractable. Simulation- based inference (SBI) methods

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sidestep likelihood evaluation by training a neural conditional density estimator qui (0 xx) (e.g. conditional normalizing flow (25)) to approximate the posterior p(0 æ.). Very often, when æ, is high-dimensional, neural statistics encoder (NSE (22)) h is used to obtain lower-dimensional representation with sufficient information for the inference task. Eventually, neural posterior estimation (NPE), one minimizes the expected negative log-likelihood over simulator draws w.r.t
parameters
CNPE(t) w) = Eo-p(00, =o- mpp(- (0) [- log qu (0) how (xx))]
Under sufficient expressiveness of the density estimation model du and the encoder hwv while considering access to arbitrarily large simulated data Dsni
[00 an NaBt the density p(0 x.))
can be approximated by sampling from q8 (0) /-- (x.)) 26 27
2.2 Transductive Semi-Supervised Posterior Estimation (RoPE)
In the presence of model misspecification, the simulator-inducee posterior p(0 xx,) may be biased relative to the true real-world posterior p(0 ær). Consequently, both NSE h and the NPE go.. when trained solely on simulated data, may fail to perform reliably real observations. RoPE 22 on which this work builds, addresses this in two stages.
NSE Fine-tuning. Once the NSE, hwr, and NPE, q from Section 2.1 are trained and fixed, a small set (calibration set) of real observations x and their corresponding known parameters 0,, is used to adapt the encoder to the domain shift. Each 0, is passed through the simulator to obtain the corresponding x S(O;) and the calibration set becomes Dealib {,t, sJ Nootoo Then, Jø. the NSE for the real observations, initialized as hw., is fine-tuned on Dealib to minimize the mean squared error between g (xx,), and h (xxt) for every pair of æ x Dealib The outcome is limited (depending on the size of Deatis) domain adaptation of the NSE gø, enabling it to encode representations of real observations in the same latent space h (X.).
Entropic OT coupling. Since NSE fine-tuning relies on limited calibration set, its ability to generalize to unseen real observations is constrained, leaving residual uncertainty in the domain transfer. RoPE takes this uncertainty into account by coupling real and simulated observations through entropic OT, which computes a soft assignment matrix between embeddings {go- (xx)) } of
test observations Diest {x+ J. and embeddings {/w- (xz of fresh simulation set Dor { The latter then acts as prototypes which the matched embeddings {ga- (art.) must be close in an Euclidean sense, similarly to the soft Kmeans algorithm (28). Specifically, they propose
solve for the following semi-balanced entropic OT problem 22 301:
P arg min Nor)) (P, C²) + PKL(P" 1 Nor _1Nor) (P, log (I)
where P is constrained row-wise to ((Nessts NOT) (P E Mient Nor Pl Neast Nieae and C is the pairwise euclidean distance matrix between both sets of embeddings. The weight p encourages prototypes to be matched to a uniform number of test samples, enabling unbalanced OT to accommodate prior misspecification, while controls the entropy regularization strength, thereby tuning the method's sensitivity to model misspecification and to uncertainty introduced during encoder fine-tuning on the calibration set. Problem is commonly solved using an iterative bregman projection solver [3 32 33). As detailed in Appendix, it comes down to actualize along iterations t the transport plan following:
B
1 N
with K(0) exp (-C- p1l Log(Nor P(UT 1T))
pi(+1) diag(
(())
Ntest K(t
Finally, (22) shows that the calibrated posterior for each æf can be approximated by marginalizing over x,, resulting in the following posterior:
Nor
0 x = Σ
aij qu- (0 hww- (xx!) with ai = Neest P.j.
(3)
3

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3 Methods
RoPE, while offering robust posterior estimation, requires computing the OT coupling over the entire test batch Drest at once, rendering the approach inherently transductive. This limits its ability to generalize to unseen observations without re-computing the transport plan.
We propose new Framework for Robust Inductive domain transfer in misspecified Simulation-Basee Inference named FRISBI. It relies on unified workflow that achieves a joint distribution-level and point-wise alignment while enabling inductive inference, thereby extending the solution to misspecified SBI beyond the limitations described above. The encoder 9. first trained using a joint objective that combines variant of entropic OT admitting closed-fore form solutions, with a supervised calibration loss, as detailed in Section 3.1] Then to avoid the need for accessing simulations at test time, we further amortize this solution using the inductive strategy presented in Section (.2 A
complete description of the full pipeline and training procedure is provided in Algorithm 3.2 For clarity, Appendix includes a summary table describing the different datasets.
3.1 Balancing Unpaired Alignment and Point-wise Domain Transfer
In addition to the data assumed to be available in RoPE [22] we also assume access to a large,
unpaired dataset of real observations, denoted as D., = { x 11=1- Furthermore, we can generate a large set of simulations, separated from the one used to train the NPE, forming the dataset Dor {x1) Nor. We propose to learn encoder Iş that optimizes joint objective, denoted Cjain. composed two terms. The first component coincides with the entropic OT objective used in RoPE (see Eq. ) with the (allumn-marginal constraint parameter fixed at p 0. It defines coupling between the encoded real samples 96 (α))) and the fixed simulated representations h (x²)) The second component operates on the calibration Dealib and aims control the deviation of the embeddings of real samples from those of their paired simulated samples. Formally, Jo is trained by solving the following problem:
arg min
Σ [Pollas - nw- (02))) P log +1 Σ EDentsh
(190(a))) hw- (x)))) (4)
a E.EDD
Entropic OT
Supervised Loss
where P E B(Nu, NOT) is optimal coupling between both distributions and (7, A) are regular- ization hyperparameters. We stress that for the supervised loss, all paired samples (xy, x.) in the calibration set Deatib taken. Whereas for the OT loss, the set D, is a combination of batch B, sampled from the unpaired dataset D., and samples from the calibration set Deatsb. while the set D, consists of the entire simulation dataset Dor well as simulated samples (æ,) from the calibration set Deatib- Intuitively, when calibration pairs are accurate (i.e., they share the exact same latent parameters 0), minimizing the supervised loss the calibration set tends to sharpen the transport supervie plan, resulting in aligoment between the OT and supervised objectives. In contrast, when calibration pairs are noisy or mismatched, the two objectives may conflict, allowing the OT term compensate for uncertainties in the calibration set.
We specifically enforce p 0 in the entropic OT objective as the resulting optimization problem w.r.t P is naturally well-suited for inductive learning. Indeed, one can see that setting = 0 in In., Eq. implies that this problem admits a closed-form solution p = diag( N.Ki K where
K C/7 and C is the pairwise euclidean distance matrix between embeddings (see also [3] Proposition 1J). This leads to the efficient stochastic gradient descent (SGD) algorithm described in Stage of Algorithm 3.2 which alternates between computing embeddings for real observations and independently computing the corresponding closed-fone form couplings for each embedding. Setting p enables this closed-form computation, requiring Nor operations per sample in D,, each involving Euclidean distance in Rd for an overall complexity of (d). In the mini-batch setting, this yields a per-step complexity of Nord, where B, is the batch size. Remark that an analogous strategy could be applied when p > 0, replacing the closed-form computation with iterative updates of Eq. until convergence, as typically done in mini-batch OT 23 24 However, this approach more computationally demanding and prone to bias, with high sensitivity to batch size. In addition,

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RoPE, which uses p > 0 and cannot operate on mini-batchess scales as Oflog(N, N Nu where N₁, is the total number of real observations. These considerations further motivate our choice of p 0 for improved efficiency and scalability.
test sample o can be directly computed by evaluating Ctest.j using the trained encoder Jør and the transport plan P*, obtained in closed form. This yields the posterior mixture as defined in RoPE (Eq. 3
Finally, once the model is trained, the posterior mixture coefficients O:j N₂. Pij for
3.2
Amortization of OT-based Posterior Estimation
Although the pipeline described in Section 3. enables inductive posterior estimation, it still relies on access to the same simulations used to train the loss in Eq. 4 at test time. To mitigate that, we propose to fit conditional normalizing flow (cNF) qs that approximates the OT-based posterior mixture, conditioned directly on the real observation embeddings Iø- (z,,)-
To fit ge, maximize its expected log density under the target mixture,
arg max Eo-prarge (0]=) [log qe (0 =)].
where Ptarget (0|z) is computed as in Eq. By the linearity of expectation, this is equivalent
aij a (0th (ar1)) [log q =)].
In practice, we approximate each inner expectation by drawing K samples {0(1)) K from qu- (0 h (x/)), and train de with the of unpaired real observations D., and simulations Dor on Lyow
]=
NOT
log q (O(..) ==)
arg min
Br
SE
During inference, for given test embedding Ia- (xtest), p(0 x,) can be approximated by sampling directly from qe- (0 z) without requiring any access to simulations.
Training Procedure
Datasets: D, = Na
No Dor (ar
NoT
Dealth = {(xt,
( Neutss
Trained Models: NSE h NPE qui
Stage 1: Joint supervised & OT Training
Stage 2: Conditional NF Amortization
1: wy +- h (xx)) Va DOT
1: Z 2, +- Io- (xx) Vxy D., exp(-1|z - /7)
Wie +- has (xt) Va Deatib for e to epochs do
2: Oki Σ, exp(- -- 1/55 Vzi, Wi
for batch B, {xf Jies, from D., do B,
3: for é = 1 to epochs do
for batch B, (ii from 2 do
2; + (xx)
Sample 0j. ~ 4- 0 w
Zic 5- 96 (xx))) Vi E Deatib
Compute L 5 Vi B,, j.
expl 1=:-- P//)
Psj Bi Σ, exp 1 1 - 1 /1)
7: Update & by gradient step
8:
Compute Gooine
Vi E B, ic,
Update @ by gradient step
Inference: z Ior (x,), p(0 xx) qe- (0
by sampling 0 (O-(0
5

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4 Experiments
4.1 Benchmarks
We evaluated our proposed approach on four benchmarks: synthetic one, two real but controlled ones, and one complex real-world benchmark. In the synthetic setting, real observations are emulated using a more complex simulator. The two controlled benchmarks involve data sampled from real systems with experimental control. The final benchmark contains real-world observations collected "in the wild," with no control over sample distributions or nuisance variable variations. The first three benchmarks were also used to evaluate RoPE [22], the main baseline method.
Pendulum. A widely-used synthetic test case in hybrid modeling and simulation- based inference literature [I3 12 [4]. The simulator models the displacement of ideal, frictionless pendulum, determined by its natural frequency wo and initial angle Φn- To emulate real observations, we use a damped pendulum model that introduces friction into the system. The damping is controlled by I friction coefficient a E Rt. The parameters we aim to infer are 0 = {wo E 10 n], do E [-x, r]}.
Causal Chambers [34]. Two real, controlled datasets collected from experimental rigs- wind tunnel and light tunnel with adjustable parameters. In the wind tunnel, the target parameter is the hatch opening angle 0 = (H e [0, 45"]). We adopt model A2C3 from [4 as the simulator, which captures pressure dynamics and hatch mechanics, while simplifying aerodynamics and omitting sensor noise, actuator delays, and environmental effects. In the light tunnel, the parameters are the RGB light intensities and a polarizer attenuation factor: 0 {R, G, B [o, 255], a E [0, I]]. We use model F3, which simulates photodiode and camera responses under varying exposure and gain, but ignores optical aberrations and sensor noise.
Real Hemodynamics Data. This benchmark uses subset 6 35 of the MIMIC-II dataset (36). comprising 350 patients who underwent thermodilution procedure estimating cardiac output (CO) via cold fluid injection and downstream temperature measurement. Each patient has arterial blood pressure (ABP) signals aligned with CO readings, yielding ~ 2200 valid ABP segments with corresponding CO values. For simulation, we use OpenBF (37]. validated ID cardiovascular flow simulator supporting fast, multiscale finite-volume simulations. The estimated parameters are (HR, CO), where HR is heart rate, obtained from ECG measurements. The empirical means and standard deviations of HR and CO in the dataset are (87, 12) beats/min and (5.1, 1.6) L/min respectively. In contrast, the CO from the simulations derived by the known connection CO = HR x SV, where SV is the stroke volume. HR and SV are sampled from uniform distributions U(50, 150) beats/min and U(40, 140) L/beat respectively. This yields CO in the range of [2, 20) L/min. This use case is inherently affected by label noise, since the pairing of HR and CO values with observed pulse waves depends on temporal alignment through patients' electronic records, which is prone to misalignments.
4.2 Experimental design
However, direct comparison on the same test set is unfair, as RoPE purely transductive. Nevertheless, we include this setting as baseline, denoted RoPE full test. For fairer inductive comparison, we introduce a single-sample variant RoPE: for each test point sess we add it to the unpaired training set D.., compute the OT coupling with simulations Dor, and estimate the posterior using the resulting plan. This is repeated per test point, and we denote this baseline as RoPE single sample. We also compare against baselines from (22). including NPE (see Section 2.I). D. applied directly to real observations without domain transfer. To assess the role of the calibration set, we also include an unsupervised domain adaptation (UDA) NPE baseline following (38]. In addition, we include OT-only basclines that do not adapt the embedding space. Here, the fixed encoder h is applied to both real and simulated samples, and OT is computed in this space. The full-test variant is denoted OT-only (full test), and the single-sample variant as OT-only (single sample). For all experiments involving OT, we use solvers from the POT
Baselines. The primary method we compare against is RoPE [22.
Python library 33 40)
Another baseline finetune-only, where the finetuned encoder gøs applied to test samples, and NPE is used to estimate the posterior directly from gø- (I,)), without OT-based mixing.

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Finally, we include two additional baselines: the upper bound SBI, where NPE is trained and tested on simulations, and the prior estimator.
Metrics. We evaluate our method using the same metrics as in 22 which assess two critical aspects of posterior estimation: accuracy and calibration.
The first metric is the log-posterior probability (LPP), defined as the average logllikelhhood of the true parameter values under the estimated posterior distribution. It measures how much density the true parameters receive in their estimated posteriors, effectively capturing the sharpness and accuracy of the model's predictions. Higher LPP indicates better match between the estimated posterior and true value.
The second metric is the average coverage area under the curve (ACAUC), which reflects how well the model's 's credible intervals align with the true parameter coverage. It provides a measure of calibration by comparing the fraction of true parameters falling within the estimated credible intervals at different confidence levels. In all experiments, credible intervals are obtained by drawing 1000 samples from the corresponding approximate posterior distribution. A perfectly calibrated model has an ACAUC of zero, while positive values indicate overconfident estimates and negative values indicate underconfident estimates. For a detailed mathematical definition, we refer the reader to 22
4.3 Results
In this section, we present the key findings of our experiments. detailed description of the implementation can be found in the supplementary material.
4.3.1 Performance Across Different Calibration Set Sizes
We evaluate methods that rely on calibration data using 5-fold cross-validation. In each fold, different, randomly sampled subset of the calibration data is used for training and validation, with independent random initialization of model weights. This approach captures both data variability and the effects of random initialization, providing robust assessment of performance.
We evaluate calibration set sizes of 10, 50, 200, and 1000 samples, while keeping the test set size fixed at 1000 samples across all benchmarks. The simulation set, DSRr, used to train both the NSE and NPE, well the unpaired real observations set, D.. each contain 1000 samples. Similarly, the set of simulations used for the OT in RoPE and our proposed joint training, Dor, also consists of 1000 samples. For training the amortised posterior estimator (Section 3.2). we exclusively use Du- Following the guidelines in (22), the entropy regularization weight 7 is set to 0.5 for all baselines involving OT, including our joint training approach. Finally, since this section focuses on the setting where both simulations and real observations are drawn from the same prior distribution p(0), we use balanced Sinkhorn OT for RoPE, effectively setting p eq. to a very high value.
The results, including mean scores and standard deviations computed across folds, are presented in Fig. Our method (solid red line) consistently outperforms the transductive single-sample RoPE baseline (solid orange line) in terms of LPP, while maintaining ACAUC close to zero across most calibration sizes. This indicates that it achieves robust inductive parameter estimation compared to RoPF. With larger calibration sets, our method matches or even exceeds the performance of the full-test RoPE (solid green line), which has access the entire test highligiggg the effectiveness of combining based domain transfer with supervised calibration. It also highlights the importance of incorporating calibration set, even small one, the UDA baseline completely fails under significant misspecification, consistent with the observations in 41 However, as the calibration set size increases, our method like the fine-tuning baselines shows decline in confidence (lower ACAUC). In contrast, the RoPE variants are less affected by this trend, suggesting that our joint training approach increasingly relies on supervised loss over the OT loss as more calibration data becomes available. We also evaluate performance moderate misspecification (Appendix [C.Jp, also considered in (22. Similar to RoPE, FRISBI shows no significant advantage over bascline methods in simpler and minimally misspecified setting.
two additional benchmarks exhibiting more
7

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Pendulum
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Figure Results across different calibration set sizes.. The top row displays performance terms of LPP (f) while the bottom one is the calibration metric ACAUC (+ 0 (-). The horizontal axis indicates the sample size while the vertical one is the metric value. Baselines that do not rely on a calibration set represented by fixed horizontal dashed lines for casier comparison.
4.3.2 Ablation
The importance of joint training, As described in Section 3 our full pipeline consists of two main components: joint distribution and point-to-point alignment, which we refer as joint training only, and the posterior-mixture amortization step. which we refer to as amortized solution only. To assess the individual contributions and necessity of these components, we evaluate their performance separately and in combination. For the amortized solution only baseline, we train the amortization step described in Section 3.2 directly to approximate the posteriors estimated by RoPE, using the unpaired real set D., and the OT simulations set Dor- For the joint training only baseline, we evaluate the posteriors obtained through our joint training approach described in Section 3.]] without the additional amortization step. Finally, we compare these to the full pipeline, which combines both components as described Section 3
The mean scores and standard deviations for each variant are reported in Fig. Overall, the joint training approach and the full pipeline achieve comparable performance in terms of LPP and ACAUC, both consistently outperforming the amortized version of the transductive RoPE solution (blue). This suggests that the performance gains in our pipeline are largely attributable to the joint training strategy. The full pipeline offers the added benefit of not requiring access to the simulations Dor test time. Notable differences arise in specific cases: the full pipeline (green) performs better in the low-calibration regime of the light tunnel benchmark (middle of Fig. 3) suggesting it may act as a regularizer for the posterior mixture when calibration data is limited, while joint training alone (orange) outperforms the full pipeline in the pendulum benchmark. In the latter, we observed that in one fold, the cNF model used to amortize the posterior mixture failed to reduce its training loss (Eq. 5), indicating difficulties in learning the posterior. This could potentially be mitigated by using more expressive cNF model.
Hyperparameter sensitivity analysis. In Appendix we present sensitivity analysis of the hyperparameters y and on the Light Tunnel benchmark. As in RoPE, a larger y beneficial when the calibration set is small, producing more diffuse OT coupling. With larger calibration sets (e.g.. 1,000 samples), y can be reduced to obtain sharper coupling. The effect of X evaluated under 10% label noise, since noise influences the weighting of the supervised objective. Higher À values decrease confidence as the calibration set grows, while smaller values yield better I.P scores, suggesting that an adaptive tuning of À based on noise level and data size may be advantagcous.
8

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Pendulum
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Figure 3: Ablation Analysis. Comparison of joint training only (..I. solution amortization only (.2), and the full pipeline. The horizontal axis shows the number of calibration samples, while the vertical axis represents the LPP(1, top) and ACAUC(- t-, bottom) scores.
4.3.3 Label Noise Robustness
In more realistic scenario, the calibration set known parameters 0 are themselves measurements and thus inherently noisy. For example, in the case of CO (cardiac output) labels, both the measurement procedure and the temporal alignment with the corresponding ABP (arterial blood pressure) signal can introduce noise. This means that the resulting simulated observations æ, may also be inaccurately paired with the real observations, potentially affecting the quality of the calibration set.
To evaluate the robustness of our proposed method label noise, we add Gaussian noise to the calibration labels, corresponding to 1% and 10% of the parameter range of the assumed prior in the light tunnel benchmark. The models clean for evaluation, as in the previous subsections. In this experiments we still consider the balanced OT settings (no prior mismatch) and keep y at 0.5. In Fig. 4 our method (solid red line) almost consistently achieves higher LPP scores than the single-sample RoPE baseline (solid orange line), indicating greater robustness to label noise due to its inductive nature. While it somewhat more sensitive to high noise levels compared to the full-test RoPE baseline (solid green line), this sensitivity diminishes with larger calibration sets. For example, with 200 calibration samples, the performance gap noticeably narrows.
trained using these noisy labels, while the
remains
In the CO estimation experiment, we explicitly account for the increased uncertainty in the calibration set by setting the entropic regularization parameter to a higher value of 1.5 for the RoPE and OT baselines and in our proposed method. Additionally, in the RoPE baselines we use unbalanced OT settings to handle known prior mismatches, as suggested in [22]. The calibration set size in these experiments is 200, and we report results across random splits assess robustness. It is important
to note that clean test-set 0 labels are not available in this setting. As shown in Fig. 5 Our method (in red) significantly outperforms all baselines, including RoPE, terms of LPP, while showing slight overconfidence (ACAUC < 0.2), highlighting the inference benefits of our amortized inductive framework real complex settings.
5 Discussion and Conclusion
In this work, we introduce an amortized and inductive posterior estimator for misspecified simulation-bage based inference. Our method leverages mini-batch optimal transport to enable joint training over both unpaired and small paired calibration sets. The final posterior, approximated by an OT-based mixture, is amortised by conditional normalizing flow, eliminating the need for additional transport computations or access to simulations at test time a key limitation of transductive approaches like RoPE. Our approach demonstrates competitive performance across

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10
% added noise
Figure 4: Label Noise Robustness. Impact of increasing label noise on performance, measured by LPP (t, top) and ACAUC (+ 0 4-, bottom), across three calibration set sizes (noted above each panel) on the light tunnel benchmark. The horizontal axis represents the noise rate, while the vertical axis shows the metric score.
QT-anlyfful test) fmotune-only
RoPElsingle sample)
SBI
CT-oniylsiggl sample)
RuPE (full test)
3.H4
I.PP
ACAUC
0.00
-2.7
18
-389
(HR,CO)
-00.0
Figure 5: Cardiac Biomarker Estimation. Performance comparison across all basclines for heart rate (HR) and cardiac output (CO) estimation, using calibration set of 200 samples. On the right, an example real and simulated arterial pulse waveforms shown.
range of benchmarks, including both synthetic and complex real-world datasets.
Dimensionality and scalability. While our experiments focus on relatively low-dimensional cases representative of many real-world scenarios, our framework naturally extends to -igher-dimensional settings. Two main challenges arise: (i) the mismatch between simulated and real embeddings tends to increase with dimensionality, requiring larger calibration sets, and (ii) the embedding dimension needed to capture sufficient statisties typically grows with the parameter dimension, complicating the coupling. Following Chen et al. [42], we use an overparameterized embedding dimension of 16 mitigate this issue. The joint optimization of supervised and OT objectives further alleviates the limitations of small calibration sets by exploiting unlabeled data. As discussed in Section 3.1 FRISBI expected to scale more efficiently than RoPE in bigh-dimensional settings.
Limitations and future directions. Sensitivity to label noise, as observed in our experiments, suggests that training data quality significantly impacts posterior accuracy. While this could be partially mitigated with higher entropy regularization weight, more adaptive joint loss formulations should be explored and specifically adaptive updates of the supervised loss weight X. Additionally, active learning or uncertainty-avee aware sampling strategies could be investigated guide calibration set selection under a fixed budget. Finally, the training process of our proposed method still involves two separate stages: the joint OT-supervised training and the subsequent amortization the induced posterior mixture. A more holistic alternative could involve learning the transport mapping directly through neural OT framework, potentially integrating the matching and inference stages into single unified process.
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A Datasets clarification
Table 1: Summary of datasets used in the pipeline.
Dataset
Observations
Labels
Usage
DSBI
simulated (æx)
yes
train NPE, NSE
Dor
simulated (æx)
train the OT objective in 3. amortization of posterior mixture in 3.2
Dealib
real (xx)
yes
train 3.1 MSE objective (and OT)
Du
real (x,)
train
3.1 OT objective, amortization of posterior mixture in 3.2
B Semi-balanced Optimal transport solvers.
We develop next how to solve for semi-balanced. entropic optimal transport problems discussed in
Sections 2.2 and 3
The overall problem reads
P arg
min PEB(N
INT... C(P) = (P, C) PKLP 1
1 Nor [Nor) +7 (P,log P). (6)
This type of problem be generally solved using iterative bregman projection solver
31 32 29 33 or equivalently mirror-descent algorithms following the KI. geometry. It comes down the following steps:
Step 1. Compute the gradient of the objective function
VL(P()) C pl log P(T + log p(t)
(7)
step 2. Then for given learning rate т, one has to solve the problem
p(+1) +
arg min (VL(P(O), P) (KKK()))))) PEB(N
(8)
we have
(VC(P()), P) (KT(())(())
= (C+ + p1 log P 1)T + + p(t) P) + lo, log p(t))
(setting
T) -C + pl log Nor
POO P(OT)) T)T, P) + + log P)
(9)
p1 No
= log
P. + Po
-KK((
e1 Noo p T
with K(00 =
Hence this problem comes down to a KL projection on the set
B(Nrest, NOT) of the Gibbs kernel K(²)
As detailed in Proposition 1 in (31) this problem admits
close-form solution detailed in Equation 2
C Additional experiments
C.1 CS and SIR benchmarks
We follow the evaluation protocol of (22), originally introduced by 4
CS. The cancer-stromal cell simulator models 2D cell growth with three Poisson rate parameters (Xey Ap, Xa). Each sample includes cell counts and the mean and maximum distance between stromal and nearest cancer cells. Misspecification induced by removing cancer cells located too close to their parent.
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Figure Minimally misspecified benchmarks (CS and SIR). The horizontal axis shows the number of calibration samples, while the vertical axis represents the LPP(t, top) and ACAUC(-> 0 <-, bottom)
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SIR. The stochastic epidemic model simulates infection and recovery dynamics with rates (B, >) Observations comprise summary statistics of infection counts, timing, and autocorrelation. Mis- specification is introduced by delaying weekend infections, adding 5% of them the following Monday.
We observe that both FRISBI (red) and RoPE underperform relative the vanilla NPE baseline (dashed green) in the SIR benchmark and achieve performance comparable to the finetuning-ons baseline (blue) in the CS case. This suggests that, under minimal misspecification, standard NPE is sufficient, and in simpler systems, even a small calibration set can adequately capture the mapping between observations and parameters.
C.2 Sensitivity analysis w.r. hyperparameters: and X
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Figure 7: Sensitivity analysis of hyperparameters ? and X. Both experiments are conducted on the Light Tunnel benchmark. The À sensitivity analysis (right) is performed under 10% label noise setting. The horizontal axis shows the number of calibration samples, while the vertical axis reports the LPP (t, top) and ACAUC (+ 0 + bottom) scores. The legend indicates the different y and X values considered and their corresponding shades.
23


Reviews:
[{"review_id": "RLxh9JbNE9", "reviewer": "Reviewer_V4WH", "summary": "This paper proposes an inductive and amortized framework for simulation-based inference\n(SBI) under model misspecification, building upon the recent RoPE method. The key\ninnovation is combining optimal transport (OT)-based distribution alignment with\nsupervised calibration in a joint training objective, followed by amortization through a\nconditional normalizing flow. This enables single-sample inference without requiring\nsimulations at test time, addressing RoPE's transductive limitations. The method is\nevaluated on synthetic and real-world benchmarks, showing competitive or superior\nperformance to RoPE while offering practical deployment advantages.", "questions": "1. In Section 3.2, am I correct in understanding that the conditional normalizing flow\n   learns an implicit mapping from real observations to parameters through the following\n   transitive relationship: (a) \\theta^{(j,k)} \\sim\n   q_{\\psi^*}(\\theta|h_{\\omega}(x_s^j)) gives parameters that likely generated\n   simulation $x_s^j$, (b) High $\\alpha_{ij}$ indicates that real observation $x_r^i$ is\n   similar to simulation $x_s^j$ in embedding space, (c) The flow $q_\\xi$ thus learns\n   that if $x_r^i \\approx x_s^j$ (high $\\alpha_{ij})$, then parameters that generated\n   $x_s^j$ are also likely for $x_r^i$?\n\n2. For the real hemodynamics data, how is the calibration dataset obtained? How are real\n   $\\theta$ values measured in practice?\n\n3. How many samples were used to compute the LPP metric? This is important to avoid bias\n   as noted in Lueckmann et al. 2021.\n\n4. How does the method scale to higher-dimensional parameter spaces? The current\n   evaluation is limited to relatively low-dimensional problems.\n\n5. Have you explored approaches to mitigate the reduced posterior confidence (ACAUC < 0)\n   observed with larger calibration sets?", "limitations": "The authors provide a thoughtful discussion of computational limitations in the\nconclusion. To further strengthen the paper, consider also highlighting earlier in the\nmanuscript:\n\n- The requirement for calibration data with true parameters, which distinguishes this\n  setting from standard SBI where only observations are available\n- Potential scalability considerations for high-dimensional parameter spaces\n- The observed trade-off between calibration set size and posterior confidence\n\nThese clarifications would help readers better understand when this approach is most\napplicable.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths\n\n**Quality:**\n\n- Strong technical contribution that addresses real limitations of existing methods\n- Thorough experimental evaluation including ablations to understand component\n  contributions\n- Competitive results across benchmarks while being more practically deployable\n- Clear acknowledgment of method limitations (e.g., reduced confidence with larger\n  calibration sets)\n\n**Clarity:**\n\n- Clear problem motivation and terminology for misspecified SBI scenarios\n- Good concise explanation of the baseline RoPE method\n- Well-structured training procedure box that clarifies the algorithm\n- Appropriate use of same benchmarks as RoPE for direct comparison\n\n**Significance:**\n\n- Addresses important practical limitations of RoPE (transductive nature, need for\n  test-time simulations)\n- Enables deployment of misspecified SBI methods in real-world scenarios requiring\n  single-sample inference\n- Demonstrates effectiveness even with small training sets (1000 samples)\n\n**Originality:**\n\n- Novel combination of joint OT-supervised training with closed-form solutions\n- Creative use of conditional normalizing flows to amortize OT-based posteriors\n- Principled approach to making transductive methods inductive\n\n### Weaknesses\n\n**Clarity:**\n\n- **Section 2.2 (lines 83-97)**: The entropic OT formulation introduces the mathematical\n  framework quite directly, which may be challenging for readers less familiar with\n  optimal transport. Consider adding a brief intuitive explanation before Equation 1,\n  perhaps describing how OT finds correspondences between real and simulated\n  observations in the embedding space to handle the domain shift.\n\n- **Section 3.2**: The amortization step involves a (for me, see question below) subtle\n  but elegant insight that could benefit from additional explanation. When sampling\n  \\theta^{(j,k)} \\sim q\\_{\\psi^*}(\\theta|h\\_{\\omega^*}(x_s^j)), it may not be\n  immediately clear to readers why these parameters sampled from simulation-conditioned\n  posteriors serve as valid training targets for real observations. It might help to\n  explicitly describe the transitive relationship: since $\\theta$ generates $x_s^j$, and\n  $x_s^j$ is matched to $x_r^i$ through OT weights, these $\\theta$ values are also\n  plausible for $x_r^i$.\n\n- **Figure 1**: While the figure shows the two-stage process, the visual flow and\n  notation could be clearer. For instance, $\\rho_f$ is not defined in the caption, and\n  the complex relationships in the cNF training are understandably difficult to capture\n  visually. Perhaps simplifying the notation or adding a legend for the different\n  symbols and arrow types could help readers navigate the diagram.\n\n- **Dataset notation**: The various datasets used throughout the paper\n ($D_{calib}$, $D_u$, $D_{OT}$, $D_{test}$)\n  serve different purposes that may not be immediately clear on first reading. A brief\n  table or list before the Training Procedure box summarizing what each dataset contains\n  and its role in the method could improve clarity - though the Training Procedure box\n  itself is already quite helpful in this regard.\n\n**Quality:**\n\n- **Scalability analysis**: The experiments focus on relatively low-dimensional\n  parameter spaces (≤4D). While these benchmarks allow direct comparison with RoPE, it\n  would be valuable to include some discussion of how the method might scale to higher\n  dimensions, or to explicitly acknowledge this as an area for future investigation.\n\n- **LPP metric details**: Following Lueckmann et al. 2021's recommendations, it would be\n  helpful to specify the number of samples used to compute the LPP metric, as this can\n  affect the reliability of the estimates.\n\n- **Hemodynamics calibration data**: The paper mentions thermodilution for obtaining\n  ground truth cardiac output values (line 179), but additional details about how these\n  measurements correspond to the ABP segments would help readers understand the data\n  collection process and implications w.r.t the limitations in other SBI use cases.\n\n**Significance:**\n\n- **Calibration data requirement**: While the need for paired data with known $\\theta$\n  is inherent to the problem setting, this requirement could be stated more prominently\n  in the introduction (similar to the RoPE paper). This would help readers immediately\n  understand that this approach targets a specific subset of SBI problems where some\n  ground truth is available.\n\n- **Incremental nature**: The contribution builds naturally on RoPE by adding\n  amortization, which is a logical and useful extension. While incremental, it addresses\n  genuine practical limitations that would prevent RoPE's deployment in many real-world\n  scenarios.", "was_revised": true, "final_justification": "I maintain my positive score as the authors have thoroughly addressed all my concerns, committing to clarify the calibration set requirements in the introduction, add sensitivity analyses for λ and γ, and improve exposition throughout. The core contribution of making RoPE inductive and amortized is valuable for practical deployment of misspecified SBI. While minor issues remain (missing CS/SIR experiments, implementation discrepancies), the authors' responsive and constructive engagement with reviewer feedback demonstrates their commitment to improving the paper.", "rebuttal": "We thank the reviewer for the thoughtful analysis of our work and address next the main concerns:\n\n## weaknesses\n\n### Clarity:\n\n> W1. Section 2.2 (lines 83-97): Adding OT background.\n\nWe will clarify in the revised version the introductive parts on OT as the reviewer suggests, specifically in the Entropic OT coupling paragraph section 2.2. (essentially explaining what we mean by soft matching)\n\n> W2. Section 3.2: Clarifying final posterior estimation and amortisation\n\nPerhaps the following explanation can make Section 3.2 more intuitive and clear. From Section 3.1 (referred to as \"joint training only\" in the ablation study on p.8), we obtain an estimated posterior density that is effectively a mixture of simulation-based posteriors, with mixture coefficients learned through the method in 3.1. What Section 3.2 introduces is a way to infer this density directly from the real observation $x_r$. The idea of using such a mixture as an approximation of the true posterior $p(\\theta \\mid x_r)$ is derived by prior work, RoPE (see Eq. (4)--(5), Section 3). The key insight from RoPE is that the OT plan approximates the joint distribution over $(x_r, x_s)$. By leveraging this approximate joint distribution, we can marginalize over $x_s$ (which becomes a discrete sum in our setting) to estimate the posterior [to be clarified in the paper Section 2.2, L97]\n\n\n> W3. Figure 1: Clarifying visualisation. \n\nThank you for these suggestions that will be taken into account to improve Figure 1 in the revised manuscript. We will clarify our workflow by illustrating more clearly the inputs/outputs, which parts are optimized and the objectives they are trained on. Additionally, we will include a visualization of the baseline vanilla NPE (and NSE), which serves as a preliminary stage for our framework on top of the figure.  \n\n> W4. Dataset notation.\n\n\nWe thank the reviewer for the helpful suggestion, we will add brief sentences in the Training procedure to describe each considered dataset. \n\n### Quality:\n\n> W5. + Q4: dimensionality and scalability analysis:\n\nThank you for this interesting remark. Indeed, we consider here only low-dimensional use cases, which are common in various real-world applications. However, the discussion regarding applicability to higher-dimensional settings is similar to that presented in the baseline RoPE work. The higher-dimensional parametric space identifiability problem could be solved by generating and training the NPE over more simulated samples. \n\nThe main limitations in higher dimensions are twofold:  \n1. The misspecification between simulated and real interactions across the different $\\theta_k$ could become more significant with the increase in dimension, increasing the complexity of the matching of embeddings of real observations with simulated ones. This requires larger sizes of calibration sets.  \n2. The embedding dimension required to encode sufficient statistics for predicting the posterior is expected to grow with the parameter dimensionality, which may complicate the coupling problem. In Chen et al. [1], they propose and validate a heuristic where the dimension of the summary statistics embedding is set to $d = 2K$, where $K$ is the dimension of $\\theta$. In our implementation, we use an overparameterized embedding dimension of 16 regardless.\n\n[1] Chen Y, Zhang D, Gutmann M, Courville A, Zhu Z. Neural approximate sufficient statistics for implicit models. Proceedings of the International Conference on Learning Representations (ICLR), 2021.\n\nHowever, we expect our framework to alleviate these limitations. Specifically, the joint training of the OT and supervised objectives could help in situations where the small calibration set is insufficient to project the representation of the real observations onto the simulated ones. This can be balanced with the unsupervised OT coupling, which can be trained on an abundance of unlabelled data.\n\nMoreover, our choice of using $\\rho=0$ benefits from a closed-form solution for the coupling, computed independently for each sample in $D_u$, involving $N_{OT}$ operations. Each operation consists of computing the Euclidean distance between a pair of embeddings in $\\mathbb{R}^d$, which is achieved in $O(d)$. Since we operate in the mini-batch OT setting we have a complexity at each step of $B_t N_{OT} d$, where $B_t$ is the batch size. In comparison, RoPE which considers $\\rho>0$ and cannot operate on mini-batches has a complexity of $O(log(N_u N_{OT})N_u N_{OT} d)$ where $N_u$ is the overall number of real observations. Hence, our approach would scale better than RoPE in these high-dimensional settings.\n\nWe notice that the points regarding dimensionality and scalability are repeatedly brought up by the reviewers, and we are planning to add a discussion of these aspects in the Conclusions and Limitations section.\n\n> W6. + Q3 LPP metric details: LPP computations and number of samples\n\nThe number of test samples for the pendulum/causal-chamber benchmarks is mentioned in lines 226-227 (1000 for all, the same scale of test set size as in RoPE). Indeed for the pulse-waves benchmark it is not explicitly mentioned, but it's also 1000. Thank you for noticing, we will add it to the revised manuscript. As for the computation of LPP, there is no need to sample from the posterior as the normalising flows model models the density function of the posterior and we just derive from it the log density value of the ground truth parameter. For the calibration score, the acauc, we sample a 1000 samples from each posterior density.    \n\n> W7. Hemodynamics calibration data: data collection & labelling process\n\nIndeed, the process of aligning the CO measurements obtained via the thermodilution method to the corresponding ABP segment is important for understanding the root cause of the noisy labels. Due to space constraints, we did not elaborate on this in the paper, but we will include a paragraph about it in the supplementary material in the revised manuscript.\n\n\n### Significance:\n\n> W8. Calibration data requirement: being more explicit about this requirement. \nWe mention the use of a calibration set and the semi-supervised settings in lines 37-39 in the intro when mentioning the prior work of RoPE. However we can make it clearer that we follow the same settings in our work. \n\n### Questions\n\n> Q1. Section 3.2, amortisation clarification. \n\n The intuition is correct. As explained in response to W2 RoPE has already showed that the real posterior can be approximated by the mixture of simulated posteriors obtained by the coupling. So indeed the $\\alpha_{ij}$ correspond to the level of \"match\" between the parameters $\\theta$ of a real observation and those of a simulated one. In 3.2 however, we amortise this mixture-based density using normalising flows. \n> Q2. For the real hemodynamics data, how is the calibration dataset obtained? How are real $\\theta$ values measured in practice?\n\nThe heart rate is measured from ECG and the cardiac output by the invasive thermodilution. Then the HR and CO measurments are aligned with the corresponding pulse-wave segment using the patient's electronical record, which can induce noisy missaligned pairs of $x$ and $\\theta$. We can elborate about that in the revised manuscript in the supp. material. \n\n> Q3. How many samples were used to compute the LPP metric? This is important to avoid bias as noted in Lueckmann et al. 2021.\n\nPlease refer to W6\n\n> Q4. How does the method scale to higher-dimensional parameter spaces? The current evaluation is limited to relatively low-dimensional problems.\nPlease refer to W5\n\n> Q5. Have you explored approaches to mitigate the reduced posterior confidence (ACAUC < 0) observed with larger calibration sets?\n\nWe believe that it is essentially a matter of hyperparameter tuning on ($\\gamma$, $\\lambda$), that we did not empirically explored. We envision solutions for future works considering for instance adaptive strategies for $\\lambda$ given $\\gamma$. One of them could be to merge both losses into a unified transport problem with additional constraints on the transport plan where $\\lambda$ would become similar to a lagrangian multiplier."}, {"review_id": "Zytxcg3g3Z", "reviewer": "Reviewer_TtJD", "summary": "The paper proposes an inductive and amortized method in the context of simulation based inference (SBI). Recent published works on SBI, RoPE operate in a transductive setting which uses optimal transport to align simulations to real data, but this needs recomputing the transport equation every for every inference time, which is a major limitation. This paper overcomes this limitation by using a joint training framework for the simulations and real data and an amortised posterior estimation. Experiments on a set of simulated and real world benchmarks provide evidence for the viability of this method.", "questions": "I mainly want some clarifications and some questions: \n\n1) To make sure I understand correctly: the supervised calibration loss is mainly useful when the calibration pairs are accurate, while the OT loss provides robustness when those pairs are noisy or mismatched. Is that correct?\n2) The evaluation datasets have only up to a maximum of four parameters. Oftentimes, complex simulations have a much larger parameter space in fields where SBIs are frequently used, such as complex fluids or astrophysics datasets. Can the authors describe how they expect their model to perform when the parameter dimensionality is very high?", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths: \n1) The paper is clearly written and most of the components going into the framework are explained reasonably well. Figures are clear as well.\n2) Addresses an important problem of simulator misspecification in SBI. This is important to sim to real problems in general.\n3) Clear motivation for improving over RoPE’s transductive limitations.\n4) This method could be practically useful since it avoids simulation access at test time which is neat. \n6) The paper provides useful ablation studies to show the importance of the joint training and amortization components.\n\nWeaknesses: \n1) An intuitive explanation of optimal transport is not provided in the main text. This can make it somewhat hard to understand for the readers.\n2) The method seems to be suspectible to noise in the calibration set, which can potentially limit its applicability in a wide range of real world problems where calibration sets are scarce and noisy.\n3) The paper uses mini-batch OT with a fixed entropy regularization parameter gamma, but does not explore how sensitive results are to this value.\n4) The method is evaluated only on problems with up to four parameters. SBI applications often involve ten or more parameters. It is unclear how the OT coupling or the normalizing flow posterior will scale in such regimes, or whether performance degrades under the curse of dimensionality.", "was_revised": true, "final_justification": "I maintain my positive assessment for the paper given the authors have satisfactorily answered most of my questions. However, I do not think the paper will have that high of an impact to the field which is what the definition of a higher rating requires. But overall, I think this is a good paper.", "rebuttal": "Thank you for the constructive review and for highlighting the practical benefits of our approach and key areas for improvement. We address the reviewer’s concerns point by point below.\n\n## Weaknesses:\n\n> W1. An intuitive explanation of optimal transport is not provided in the main text. This can make it somewhat hard to understand for the readers.\n\nThank you for the suggestion, we will clarify the motivation and operations involved in Equation 1 in the corresponding section of the revised manuscript.\n\n\n> W2. The method seems to be suspectible to noise in the calibration set, which can potentially limit its applicability in a wide range of real world problems where calibration sets are scarce and noisy.\n\nIt is true that we have identified some sensitivity to noise in the synthetic noise experiment. However, our approach worked better than the RoPE-single-sample baseline, and since RoPE-full-test is calculated and evaluated on the test set itself, it is perhaps able to couple even noisy samples when the noise model is relatively simple.\n\nIt should be emphasized that in the real noisy benchmark (the cardiac biomarkers), our method outperformed RoPE. For clarity, we will emphasize this point in the revised manuscript in the results discussion.\n \n\n\n> W3. The paper uses mini-batch OT with a fixed entropy regularization parameter gamma, but does not explore how sensitive results are to this value.\n\nWe performed a sensitivity analysis w.r.t the entropy regularisation param $\\gamma$ on the light tunnel benchmark. The results below are for the lpp metric. \n\n\n| num_samples (calib.) ↓ / γ → | 0.01           | 0.1            | 0.5            | 1              | 5              |\n|---------------------|----------------|----------------|----------------|----------------|----------------|\n| 10                  | -8.83 ± 2.92   | -6.83 ± 2.19   | -2.57 ± 1.04   | -1.85 ± 0.69   | -2.27 ± 0.85   |\n| 50                  | -0.98 ± 0.60   | -0.54 ± 0.41   |  0.09 ± 0.28   |  0.02 ± 0.12   | -1.40 ± 0.06   |\n| 200                 |  0.13 ± 0.24   |  0.47 ± 0.17   |  0.66 ± 0.03   |  0.37 ± 0.04   | -1.27 ± 0.07   |\n| 1000                |  1.00 ± 0.08   |  1.15 ± 0.06   |  0.96 ± 0.05   |  0.54 ± 0.03   | -1.22 ± 0.06   |\n\nLike in RoPE, we observe for our method that when the calibration set is small, it is beneficial to select a larger value for the entropy regularization parameter, as the OT matching becomes more diffuse i.e closer to an uniform distribution. On the other hand, when the calibration set is large (e.g., 1000 samples), it is more likely that a good pairing exist between those and the real data, hence we can reduce $\\gamma$ to obtain a sharper coupling.\n\n\n\n\n> W4. + Q2 The method is evaluated only on problems with up to four parameters. SBI applications often involve ten or more parameters. It is unclear how the OT coupling or the normalizing flow posterior will scale in such regimes, or whether performance degrades under the curse of dimensionality.\n\nIndeed, we consider here only low-dimensional use cases, which are common in various real-world applications. However, the discussion regarding applicability to higher-dimensional settings is similar to that presented in the baseline RoPE work. The higher-dimensional parametric space identifiability problem could be solved by generating and training the NPE over more simulated samples. \n\nThe main limitations in higher dimensions are twofold:  \n1. The misspecification between simulated and real interactions across the different $\\theta_k$ could become more significant with the increase in dimension, increasing the complexity of the matching of embeddings of real observations with simulated ones. This requires larger sizes of calibration sets.  \n2. The embedding dimension required to encode sufficient statistics for predicting the posterior is expected to grow with the parameter dimensionality, which may complicate the coupling problem. In Chen et al. [1], they propose and validate a heuristic where the dimension of the summary statistics embedding is set to $d = 2K$, where $K$ is the dimension of $\\theta$. In our implementation, we use an overparameterized embedding dimension of 16 regardless. \n\n[1] Chen Y, Zhang D, Gutmann M, Courville A, Zhu Z. Neural approximate sufficient statistics for implicit models. Proceedings of the International Conference on Learning Representations (ICLR), 2021.\n\nHowever, we expect our framework to alleviate these limitations. Specifically, the joint training of the OT and supervised objectives could help in situations where the small calibration set is insufficient to project the representation of the real observations onto the simulated ones. This can be balanced with the unsupervised OT coupling, which can be trained on an abundance of unlabelled data.\n\nMoreover, our choice of using $\\rho=0$ benefits from a closed-form solution for the coupling, computed independently for each sample in $D_u$, involving $N_{OT}$ operations. Each operation consists of computing the Euclidean distance between a pair of embeddings in $\\mathbb{R}^d$, which is achieved in $O(d)$. Since we operate in the mini-batch OT setting we have a complexity at each step of $B_t N_{OT} d$, where $B_t$ is the batch size. In comparison, RoPE which considers $\\rho>0$ and cannot operate on mini-batches has a complexity of $O(log(N_u N_{OT})N_u N_{OT} d)$ where $N_u$ is the overall number of real observations. Hence, our approach would scale better than RoPE in these high-dimensional settings.\n\nWe notice that the points regarding dimensionality and scalability are repeatedly brought up by the reviewers, and we are planning to add a discussion of these aspects in the Conclusions and Limitations section.\n\n\n\n## Questions\n\n\n> Q1. To make sure I understand correctly: the supervised calibration loss is mainly useful when the calibration pairs are accurate, while the OT loss provides robustness when those pairs are noisy or mismatched. Is that correct?\n\nIndeed, that is the case, but not only. Recall that the main setting we consider here is one where the calibration set is too small for a purely supervised domain transfer. To address this, we introduce an additional unsupervised OT loss. The OT loss provides a regularisation for matching real observations with simulated ones, even when the calibration set is small, which is the basic setting of the problem we aim to solve.\n\nThis effect is evident from the baseline experiments in Figure 2, where we observe that our method outperforms the finetune-only baselines at small calibration set sizes. However, as the size of the calibration set increases, the performance gap diminishes.\n\n\n> Q2. dimensionality and scalability\n\nPlease refer to W4"}, {"review_id": "qS8QBHBCs1", "reviewer": "Reviewer_wUUr", "summary": "This paper deals with the issue of misspecification in amortized simulation-based inference (SBI).\nSpecifically, in amortized SBI such as neural posterior estimation (NPE) one would train a neural network and possibly a neural statistics encoder such that given a dataset $x$ and encoded dataset $h(x)$ --> the network learns to map to the posterior $q(\\theta| h(x))$.\nThe issue is that $h(x)$ and $q(\\cdot)$ are trained using synthetic data (via a simulator). The real generative process may deviate from the simulator, leading to model misspecification / covariate shift.\n\nPrevious work (RoPE) addressed the issue via optimal transport, but had strong requirements (e.g., requiring a batch of test samples for inference). This work addresses multiple limitations of RoPE, including provided an amortized posterior estimation method based on optimal transport. The paper first introduces the method for joint supervised & OT training, and then how to amortize it.\n\nThe technique is evaluated on four benchmarks, with additional ablation studies and an analysis of robustness to noise.", "questions": "Please see above for my open comments and related questions.\nI'd be happy to increase my score if the authors address my points.", "limitations": "The paper could be more upfront about its limitations -- naturally it inherits limitations of NPE, but it should be added which additional limitations are brought in by the specific methodology.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Overall, this is a solid paper tackling a very timely and significant issue, which is robustness of amortized SBI techniques to model misspecification (also known as *sim2real gap*). The approach is original and using state-of-the-art techniques, such as optimal transport (OT). The paper is generally well written and clear.\n\nOne potential weakness of the paper is that all considered baselines focus on OT approaches which require a \"calibration dataset\" (except for the NPE baseline) which consists of real data samples together with their true parameters. Obtaining such data for some applications may be a strong or costly requirement. \nGiven this additional constraint, it'd be useful for the practitioners to see how the proposed method compares to *robust* SBI techniques (beyond vanilla approaches) that do not require real data for calibration (see e.g. Huang et al., \"Learning Robust Statistics for Simulation-based Inference under Model Misspecification\", 2023). I would expect the paper's method to be better, since it leverages this additional information from the calibration set, but it'd be good to indeed see that this is the case.\n\nAs a minor exposition point, I recommend to state clearly the dimensions involved in each problem; that is parameter dimensionality (I think it's 2-4-1-4 or similar) and the dimensionality of the summary statistics. Relatedly, it'd be useful to know to which dimension (of parameters and data) is the method expected to scale.", "was_revised": true, "final_justification": "The authors have addressed my concerns and after considering the whole discussion and rebuttal I increased my score.", "rebuttal": "Thank you for the constructive and encouraging review, and for highlighting both the strengths of our approach and valuable directions for comparison and clarification. We address the reviewer’s comments and suggestions below.\n\n> W1. comparison with non-calibration set baselines\n\nThis is indeed a very important point. In the ICML paper of the baseline RoPE work, the authors report the performance of two baselines without a calibration set: NPE-RS (Huang et al.) and NNPE (Ward et al., \"Robust Neural Posterior Estimation and Statistical Model Criticism\"). They show that while these methods perform better than vanilla NPE, they still fall short compared to baselines that incorporate a calibration set — even when the calibration set is small. We will add this to the discussion of the baseline experiment results (Section 4.3.1).\n\n\n> W2. As a minor exposition point, I recommend to state clearly the dimensions involved in each problem; that is parameter dimensionality (I think it's 2-4-1-4 or similar) and the dimensionality of the summary statistics. Relatedly, it'd be useful to know to which dimension (of parameters and data) is the method expected to scale.\n\nThank you for the comment. We will clarify this point in the revised manuscript. In addition, we will include a discussion on scalability with respect to dimensionality. Indeed, we consider here only low-dimensional use cases, which are common in various real-world applications. However, the discussion regarding applicability to higher-dimensional settings is similar to that presented in the baseline RoPE work. The higher-dimensional parametric space identifiability problem could be solved by generating and training the NPE over more simulated samples. \n\nThe main limitations in higher dimensions are twofold:  \n1. The misspecification between simulated and real interactions across the different $\\theta_k$ could become more significant with the increase in dimension, increasing the complexity of the matching of embeddings of real observations with simulated ones. This requires larger sizes of calibration sets.  \n2. The embedding dimension required to encode sufficient statistics for predicting the posterior is expected to grow with the parameter dimensionality, which may complicate the coupling problem. In Chen et al. [1], they propose and validate a heuristic where the dimension of the summary statistics embedding is set to $d = 2K$, where $K$ is the dimension of $\\theta$. In our implementation, we use an overparameterized embedding dimension of 16 regardless.\n\n[1] Chen Y, Zhang D, Gutmann M, Courville A, Zhu Z. Neural approximate sufficient statistics for implicit models. Proceedings of the International Conference on Learning Representations (ICLR), 2021.\n\nHowever, we expect our framework to alleviate these limitations. Specifically, the joint training of the OT and supervised objectives could help in situations where the small calibration set is insufficient to project the representation of the real observations onto the simulated ones. This can be balanced with the unsupervised OT coupling, which can be trained on an abundance of unlabelled data.\n\nMoreover, our choice of using $\\rho=0$ benefits from a closed-form solution for the coupling, computed independently for each sample in $D_u$, involving $N_{OT}$ operations. Each operation consists of computing the Euclidean distance between a pair of embeddings in $\\mathbb{R}^d$, which is achieved in $O(d)$. Since we operate in the mini-batch OT setting we have a complexity at each step of $B_t N_{OT} d$, where $B_t$ is the batch size. In comparison, RoPE which considers $\\rho>0$ and cannot operate on mini-batches has a complexity of $O(log(N_u N_{OT})N_u N_{OT} d)$ where $N_u$ is the overall number of real observations. Hence, our approach would scale better than RoPE in these high-dimensional settings."}, {"review_id": "MQiHJoywXp", "reviewer": "Reviewer_jspK", "summary": "This paper introduces a method for addressing misspecification in SBI. The method builds on RoPE, a recently published method that addresses the same challenge, and eliminates RoPE's need for a calibration set of real observations during inference. This is achieved by 1.) incorporating the callibration set into the fine-tuning objective of the pretrained NPE embedding network, and 2.) by adapting the optimal transport objective to allow for inductive inference requiring only a single observation. Finally, the authors propose to train an additional normalizing flow to amortize the computation of the inference-time OT plan.\nThe authors evaluate their method in comparison to RoPE and various ablations mainly using the tasks, metrics, and experiments considered in the RoPE paper.", "questions": "- Can the authors provide some mathematical justification why the $\\rho=0$ transport plan is a good choice, and what posterior one obtains in the end? Does the self-calibration property of RoPE or something similar still hold when $\\rho=0$?\n- How should users tune $\\lambda$?\n- Have the authors considered different types of regularization other than the OT term in Eq. (4)? Replacing or dropping it would eliminate the dependence on the callibration set of observations during finetuning. How exactly does the OT term compensate, if the supervised term is off?\n- Since their paper is so closely related to RoPE, could the authors also apply their method to the CS and SIR task? Or was there a good reason not to?\n- Some numbers that should be consistent between this paper and the RoPE paper seem different? Why is that? E.g. the LPP for the SBI upper bound on the light tunnel task.", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "This paper tackles the important challenge of handling simulator misspecification in SBI. \n\n## Strengths\n- The method addresses some proper limitations of RoPE, particularly its dependence on a calibration set of real observations during inference.\n- The authors consider many ablations and include an experiment concerning label noise robustness. This is highly relevant in the RoPE setting, where access to pairs of real observations and ground truth parameter measurements (which might be noisy) is assumed.\n\n## Weaknesses\n\n### OT with $\\rho = 0$:\nTo make their method inductive rather than transductive, the authors drop the column-marginal constraint of the semi-balanced OT objective used in RoPE. This enables them to obtain a closed-form transport plan that does not depend on other observations. In particular, they write:\n> While being more computationally intensive for training, such approaches are also known for leading to biased OT solutions and a significant sensitivity to the chosen batch size, hence further motivate our choice for $\\rho = 0$.\n\nHowever, the claim about biased OT solutions and the general OT framing is slightly baffling, as completely dropping the column-marginal constraint (which is $\\rho=\\infty$ for balanced OT) results in something that basically ceases to be optimal transport?\nRather, the \"transport plan\" the authors obtain with $\\rho = 0$ is a normalized squared exponential kernel that weights certain simulations higher than others based on the distance between their embedding and the observation's embedding. \n\nThis might be a reasonable thing to do, but the authors should more clearly describe that this is what they do, and explain why this might be a reasonable thing to do.\n\n### Finetuning objective:\nThe $\\rho =0$ transport plan is also used in the finetuning objective, combined with the supervised loss used in RoPE. However, it's unclear what the exact role of the OT part in the objective is. The authors write:\n> In contrast, when calibration pairs are noisy or mismatched, the two objectives may conflict, allowing the term to compensate for uncertainties in the calibration set.\n\nThe paper does not discuss how this compensation works, especially given the special transport plan.\n\n### Complexity and hyperparameter tuning:\nWhile the authors mention this limitation, the method includes quite a few complex steps, which hinders applicability. \nUsers need to first train an NPE flow and an embedding net. Then, the finetuning of the embedding net is based on a modified objective, which introduces another hyperparameter $\\lambda$ (beyond the hyperparameter for the entropy regularization $\\gamma$).\nFinetuning is followed by training another flow to amortize over the computation of the transport plan.\n\nThe discussion of how to choose these hyperparameters, particularly $\\lambda$, and their effect on inference is insufficient.\nFurthermore the method still assumes access to two different calibration sets. \n\n### Experiments:\nWhile the authors consider many different ablations, the paper does not consider all the tasks performed in RoPE.", "was_revised": true, "final_justification": "The authors satisfactorily addressed many of my concerns, so I am raising my score from 3 to 4. Overall, this paper brings modest improvements over RoPE, both performance-wise and from a conceptual point of view (transductive vs. inductive). However, similar to RoPE, the method's dependence on a calibration set and its overall complexity, requiring many training and tuning steps, are key limitations that restrict applicability and impact.", "rebuttal": "Thank you for the constructive and detailed review, and for raising important points to clarify in our work. Below, we address them point by point.\n\n> W1+Q1. **OT with $\\rho=0$**\n\nThank you for this interesting analysis. First, we would like to clarify that the inductiveness of our approach is not inherently due to our choice of $\\rho=0$, but rather from how we optimize $g_{\\phi}$ (Stage 1) and $q_{\\xi}$ (Stage 2), allowing inference without solving an OT problem at test time. Our learning scheme can naturally be extended to $\\rho > 0$ by computing an optimal $P_{ij}$ in Stage 1 (Line 7) using RoPE’s OT solver, and integrating the computation of $\\alpha_{ij}$, and thus the OT plan, by batch for the training of Stage 2. This process remains inductive. We will clarify this in L129–140.\n\nSecond, we believe that one commonly used term for the matching problem in Equation 1 is the one used in L88: \"semi-balanced entropic OT problem\" (though “semi-relaxed” or “semi-unbalanced” are also used), following recent works like [30], [A, B] with $\\rho = 0$ and [C] with $\\rho > 0$. This corresponds to a specific instance of unbalanced OT, with $\\infty$ as hyperparameter on the first marginal and $\\rho$ on the second. After the introduction, we simplified terminology to \"OT\" or \"OT coupling\" for conciseness. If the reviewer finds it important to be more specific throughout the text, we welcome precise suggestions.\n\nFinally, we agree with the reviewer’s point on behavior with $\\rho = 0$. [D] offers good intuitions on the semi-balanced OT loss without entropic regularization. They show that minimizing the support and masses of the target distribution for a fixed input one (i.e., finding a minimal Wasserstein estimator) generalizes k-means depending on the inner cost. Entropic regularization on $\\mathbf{P}$ naturally yields a soft k-means. These problems still differ from ours, as the target support is $\\{ h_{w^\\star}(x_s^j) \\}$, and we optimize $g_{\\phi}$ over inputs $\\{x_r^i\\}$. Therefore, embeddings from simulations act as prototypes (or clusters) over which we fit real sample embeddings. Setting $\\rho > 0$ would encourage uniform use of these prototypes for fitting real data (whose validity is questionable in mini-batch settings). However, as mentioned, $\\rho = 0$ scales better than $\\rho > 0$ thanks to the closed-form; the OT solution does not depend on the sampled batch and provides empirically strong performance. We will clarify this analysis and the referenced arguments in the revised manuscript.\n\n[A] Clark, R. A., Needham, T., & Weighill, T. (2025, April). Generalized dimension reduction using semi-relaxed Gromov-Wasserstein distance. In Proceedings of the AAAI Conference on Artificial Intelligence (Vol. 39, No. 15, pp. 16082–16090).\n\n[B] Van Assel & al. (2025). Distributional Reduction: Unifying Dimensionality Reduction and Clustering with Gromov-Wasserstein. Transactions on Machine Learning Research.\n\n[C] Van Assel, H., & Balestriero, R. (2024). A graph matching approach to balanced data sub-sampling for self-supervised learning. In NeurIPS 2024 Workshop: Self-Supervised Learning—Theory and Practice.\n\n[D] Canas, G., & Rosasco, L. (2012). Learning probability measures with respect to optimal transport metrics. Advances in Neural Information Processing Systems, 25.\n\n\n> W2. **Finetuning objective**: clarifying joint OT+supervised objective\n\nThe idea was that, unlike in RoPE where the finetuning is done first, then $g$ is fixed and the OT coupling is applied to the embeddings obtained by $g$, in our framework we propose to train $g$ jointly on both the supervised objective (using the calibration set) and the OT objective (using both the calibration set and a large unsupervised set of real observations). The motivation is that, since the calibration set is small and we want to avoid overfitting by the encoder $g$, the OT objective act as a regularizer—especially given the large number of unpaired real observations it is trained on. We will make the quoted sentence clearer in the revised version.\n \n \n\n> W3. + Q2. **Complexity and hyperparameter tuning:** \n\n- We would like to clarify that the NPE training stage, followed by the finetuning of $g$, is shared between RoPE and our method. The only additional step we introduce is the amortization stage, which is specifically designed to remove the need for simulation access at test time, a requirement that remains in RoPE, even in its \"inductive\" variant (RoPE-single sample).\n\n- Hyperparameter selection: In this regard, our approach does not differ significantly from RoPE. In their work, the authors suggest that practitioners, if possible, hold out a validation set of labeled real observations from the calibration set to tune $\\gamma$ (and $\\rho$), or otherwise rely on default values informed by a sensitivity analysis. Similarly, we provide a sensitivity analysis with respect to $\\gamma$ (please refer to W3 in the reply to reviewer TtjD ) and $\\lambda$ on the light tunnel benchmark (w.r.t LPP). \n\n\n| num_samples in calib. set ↓ / λ → | 0.01            | 0.5             | 1.0             | 5.0             | 50              |\n|---------------------|------------------|------------------|------------------|------------------|------------------|\n| 10                  | -23.12 ± 3.45    | -9.70 ± 1.59     | -7.35 ± 0.94     | -7.04 ± 1.36     | -6.93 ± 1.48     |\n| 50                  | -19.06 ± 3.76    | -5.36 ± 1.47     | -4.72 ± 0.83     | -4.30 ± 0.73     | -4.28 ± 0.68     |\n| 200                 | -20.67 ± 2.72    | -3.67 ± 0.97     | -3.13 ± 0.27     | -3.06 ± 0.15     | -3.19 ± 0.32     |\n\nThe $\\lambda$ parameter was analyzed on the light tunnel benchmark with synthetic noise added at a 10% level. We observe that as long as $\\lambda > 1$, the results remain relatively stable.\n\n- Calibration set: When we refer to the calibration set, we mean the small set of labeled real observations $(x_r, \\theta)$. In both our framework and the RoPE baselines, there is only one calibration set, which is used to train the supervised objective. We kindly ask the reviewer to point us to the source of this confusion so that we can clarify it appropriately in the revised version.\n \nQuestions:\n\n> Q1. $\\rho=0$ discussion, posterior computation, self-calibration.\n- For the discussion about the choice of $\\rho=0$ please refer to W1.\n- The posterior approximation in Stage 1 for $\\rho = 0$ remains unchanged and is expressed as a mixture of simulated posteriors, weighted by the optimal transport plan coefficients: $\\tilde{p}(\\theta \\mid x_r^i) := \\sum_{j=1}^{N_{\\text{OT}}} \\alpha_{ij} \\, q_{\\psi^\\star}(\\theta \\mid h_{\\omega^\\star}(x_s^j))$, where $\\alpha_{ij} := N_u P_{ij}^\\star$. Setting $\\rho = 0$ does not alter the assumptions underlying RoPE’s equation (Eq. 4): \n$p(\\theta \\mid \\mathbf{x}_r) = \\int p(\\theta \\mid \\mathbf{x}_s) \\pi^{\\star}(\\mathbf{x}_s \\mid \\mathbf{x}_r) d\\mathbf{x}_s$, but only changes how the distribution $\\pi^\\star$ is approximated.\n- Indeed, we cannot guarantee the self-calibration property as in RoPE (balanced RoPE without the relaxation of the column-marginal constraint). Since in our case we don't constrain the column-marginal at all, we cannot guarantee that $\\sum_{i=1}^{N_u} P_{ij}^\\star = \\frac{1}{N_{OT}}$, so the derivation of the self-calibration property of balanced RoPE (RoPE appendix, p. 20) does not hold. However, in practice RoPE also relaxes this constraint, so self-calibration is not guaranteed as well. \n\n\n>Q2. How should users tune $\\lambda$ ? \n\nPlease refer to W3. \n\n>Q3. OT+supervised terms purpose\n\nThe calibration set $D_{calib}$ is a small supervised set of pairs $(\\theta, x_r)$, from which we can simulate the corresponding $x_s = S(\\theta)$ (with $S$ being the simulator). It is used to train the supervised objective, specifically, to finetune the encoder $g$ so that it embeds $x_r$ as close as possible to its corresponding $x_s$.\nThe entropic OT term compensates for the uncertainty in the supervised matching, which arises due to the limited size of the supervised calibration set. It enables a soft matching between $g(x_r)$ and $h(x_s)$, and also allows us to train on a large unlabeled set of real observations $x_r$, denoted as $D_{OT}$. As a result, $g$ is optimized jointly on both the supervised and OT objectives.\nWe observe that the pure OT baselines perform worse than both RoPE and our proposed method. Additionally, in the $\\lambda$ sensitivity experiment shown above, we demonstrate that heavily down-weighting the supervised loss (e.g., $\\lambda = 0.01$) leads to significantly degraded performance.\n\n>Q4. CS and SIR tasks\n\nWe chose not to analyze the CS or SIR tasks, as the RoPE paper indicated that the level of misspecification in these cases was limited and vanilla NPE or OT without a calibration set already performed well. Instead, we introduced a more complex use case: the real cardiac biomarker inference task from the arterial pressure wave benchmark. This represents a substantially different challenge compared to the RoPE cardiac biomarkers task, which was entirely synthetic.\n\n>Q5.  results consistency \n\nWe did not have access to the RoPE code, so we implemented our own version. Naturally, this may lead to some variations in results. However, the main reason for the differences in baseline scores is the use of batch normalization.\nAs noted in the implementation details (p. 22–25, appendix) of the RoPE paper, they do not use batch normalization in the NSE encoders, which results in a very low LPP for the vanilla NPE baseline in the highly misspecified benchmarks. To ensure a fair comparison, we added batch normalization to all baselines in our implementation (we will mention it in the revised manuscript)."}]

Author rebuttal:

Meta-review: The paper introduces an inductive and amortized framework for simulation-based inference that integrates calibration and optimal-transport-based alignment into a single trainable pipeline, enabling efficient inference without simulation access at test time and showing strong performance across synthetic and real-world benchmarks. Reviewers generally found the contribution technically solid, practically relevant, and an important extension of RoPE, though they noted that the contribution is somewhat incremental. Sentiment across reviewers is positive overall, with consensus that the method improves applicability of SBI but with concerns about clarity, scalability to higher dimensions, sensitivity to noisy calibration data, and missing benchmark experiments. To further strengthen the work, the authors should candidly and prominently discuss limitations, including the dependence on calibration data with known parameters, potential scalability issues in high-dimensional settings, and trade-offs in posterior confidence with larger calibration sets.
Decision: Accept (poster)