Neural Conditional Transport Maps
Authors: Carlos Rodriguez-Pardo
Leonardo Chiani
Emanuele Borgonovo
Massimo Tavoni
Venue: Accepted by TMLR ()

Abstract: We present a neural framework for learning conditional optimal transport (OT) maps between probability distributions. Conditional OT maps are transformations that adapt based on auxiliary variables, such as labels, time indices, or other parameters. They are essential for applications ranging from generative modeling to uncertainty quantification of black-box models. However, existing methods for generating conditional OT maps face significant limitations: input convex neural networks (ICNNs) impose severe architectural constraints that limit expressivity. At the same time, simpler conditioning strategies, such as concatenation, fail to model fundamentally different transport behaviors across conditions. Our approach introduces a conditioning mechanism capable of simultaneously processing both categorical and continuous conditioning variables, using learnable embeddings and positional encoding. At the core of our method lies a hypernetwork that generates transport layer parameters based on these inputs, creating adaptive mappings that outperform simpler conditioning methods. We showcase the framework's practical impact through applications to global sensitivity analysis, enabling efficient computation of OT-based sensitivity indices for complex black-box models. This work advances the state-of-the-art in conditional optimal transport, enabling broader application of optimal transport principles to complex, high-dimensional domains such as generative modeling, black-box model explainability, and scientific computing.

Neural Conditional Transport Maps
Carlos Rodriguee-PPardo
Deportment of Management, Economics and Industrial Engineering, Politecnico di Milano RPF-CMCC European Institute on Economics and the Environment (EIEE) Center on Climate Change (CMCC)
Leonardo Chiani
leonardo. cennarrdo.chian@@cmme..
Department of Management, Economics and Industrial Engineering, Politecnico di Milano RPF-CMCC European Institute Economics and the Ervironment (EIEE)
Euro- Ceto-Meittrranen Center on Climate Change (CMCC)
Emanuele Borgonovo
emanucle. mmanauee..orgooooo2f@maaacoo
Bocconi University, Department of Decision Sciences MIT, Department of Nuclear Science and Engineering
Massimo Tavoni
massimo..
Department of Management, Economics and Industrial Engineering, Politecnico di Milano RFFF-MCC European Institute Economics and the Environment (EIEE) Faro-Medte Mettereattennnnenn Center on Climate Change (CMCC)
Abstract
We present a neural framework for learning conditional optimal transport (OT) maps between probability distributions. Conditional OT maps are transformations that adapt based auxiliary variables, such labels, time indices, other parameters. They essential for applications ranging from generative modeling to uncertainty quantification of black-box models. However, existing methods for generating conditional OT maps face significant limitations: input convex neural networks (ICNNs) impose severe architectural constraints that limit expressivity. At the same time, simpler conditioning strategies, such as concatenation, fail to model fundamentally different transport behaviors across conditions. Our approach introduces a conditioning mechanism capable of simultaneously processing both categorical and continuous conditioning variables, using learnable embeddings and positional encoding. At the core of our method lies hypernetwork that generates transport layer parameters based on these inputs, creating adaptive mappings that outperform simpler conditioning methods. We showcase the framework's 's practical impact through applications to global sensitivity analysis, enabling efficient computation of OT-based sensitivity indices for complex black-box models. This work advances the state-of-the-at transport, enabling broader application of optimal transport principles to complex, high- dimensional domains such generative modeling, black-box model explainability, and scientific computing.
conditional optimal
1 Introduction
Optimal transport (OT) is a powerful mathematical framework for comparing and transforming probability distributions, with wide applications across machine learning, computer vision, and scientific computing. OT provides natural geometry for distribution spaces, offering stronger thcoretical guaranteos compared alternative measures (Peyré & Cuturi 2019). The importance of OT in machine learning has grown significantly, particularly in generative modeling where forms the foundation of flow matching (Lipman et al.

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and rectified flows (Liu et al. 2023). Despite its theoretical appeal and practical success, applying OT to high-dimensional, real-world problems has long been constrained by computational limitations. Classical OT methods scale poorly with sample size. Neural approaches have made significant progress in addressing these challonges by approximating transport maps with neural networks (Arjovsky et al.) 2017 Makkuva
2023
et al. 2020: Korotin et al. 2023), enabling efficient computation in high-dimensional spaces.
The optimal mapping between distributions and the related cost are two elements of interest in the OT framework. The optimal mapping is used in different fields, ranging from computer vision (Bonneel & Digne, 2023) to biology (Bunne et al. 2023). However, many practical applications require conditional optimal maps: transformations that adapt based on auxiliary variables such as labels, time indices, or other parameters. Conditional transformations in general are crucial across diverse domains: generative models need to produce samples conditioned on class labels or continuous attributes, domain adaptation requires transport maps that vary with domain-specific characteristics, and climate-cconony models need to model how distributions of climate variables change based on emissions or policy scenarios. The challenge lies in efficiently computing these conditional optimal maps, particularly in data-intensive problems where traditional OT methods become computationally prohibitive.
The OT cost has recently been presented as a valuable tool for global sensitivity analysis (GSA, Wiesel (2022); |: Borgonovo et al. (2024)). This application exemplifies the computational challenge for conditional OT. GSA leverages statistical methods to quantify how uncertainty in model outputs can be attributed to different input sources (Saltelli 2002). GSA is therefore useful for understanding complex black-box models in machine learning, climate science, and economics (Saltelli et al., 2020). Recent works leverage OT costs to define sensitivity indices with valuable theoretícal properties (Wiesel 2022 Borgonovo et al., 2024 However, these methods face a severe computational bottleneck: computing sensitivity indices based OT for complex models with many imputs requires solving hundreds to thousands of OT problems. The classical approach becomes computationally intractable due to memory requirements for storing cost matrices and the combinatorial explosion of required OT solvers. Efficiently learning conditional transport maps provides new tools for large-scale black-box model explainability.
Existing neural conditional OT methods suffer from fundamental limitations that prevent deployment in general cases (Bunne et al., 2022: Wang et al.) 2025). The first approach is based on input convex neural networks (ICNNs), imposing severe prchitectural constraints, requiring non-negative weights, limited activation functions, and specialized initialization, leading to reduced expressivity and optimization difficulties, particularly for complex conditional problems (Korotin et al. 2021). Moreover, using concatenation as a conditioning strategy fails to model complex transport patterns across conditions. This type of conditioning lacks the flexibility to learn distinct. mappings when different conditions require qualitatively different. transformations, limiting its applicability to real-world scenarios where conditioning variables significantly alter the underlying transport structure. The second approach, based on conditional OT flows, alleviates some of these expressivity issues by conditioning the velocity field of an ODE on auxiliary variables, allowing the entire transport trajectory to depend on the condition. However, this comes at the cost of more complex training: solving ODE constraints requires additional regularization terms, hyperparameter tuning is delicate, and optimization is significantly more expensive, making it difficult to scale to large datasets or high-dimensional problems. Finally, [aan et al. (2025) restricts attention to simple tractable input distributions, while we consider the general setting where input distributions are arbitrary.
In this paper, we expand the state-of-the-art neural OT framework (Korotin et al. 2023 to efficiently learn conditional OT maps across both categorical and continuous conditioning variables. Our approach leverages a hypernetwork architecture-a neural network that dynamically generates the parameters for transport layers based on conditioning inputs. This mechanism creates adaptive mappings outperforming simpler conditioning methods while maintaining computational efficiency. Our contributions are as follows:
We extend the Neural Optimal Transport (NOT) framework to conditional sottings.
We introduce unified conditioning mechanism capable of simultancously processing both categorical and continuous variables, using learnable embeddings and positional encoding.
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After testing variety of possible alternatives, we propose hypernetwork- based architecture that generates condition-specifie transformation parameters, enabling fundamentally different mappings for each condition value without the computational overhead of separate networks.
We provide empirical validation across synthetic datasets, image generation tasks, climate data, and integrated assossment modcls.
We demonstrate practical applications to global sensitivity analysis, enabling efficient computation of OT-based sensitivity indices for complex black-box models at scale. This is the first comparison of the neural OT framework by Korotin et al. (2023) with well-established solvers for this task.
We release our data and open-source implementation of our method in this Github repository
The remainder of this paper is organized follows. Section reviews related work on traditional and neural OT, and conditioning methods. Section 3 details our conditional neural transport framework, including problem formulation, architecture, and training procedure. Section 4 presents the applications and experimental sctup, while Section 5 shows results on benchmark datasets and ablations. Finally, Section 6 discusses limitations and future directions.
2 Background
2.1 Optimal Transport Foundations and Modern Applications
Optimal transport (OT) has emerged as a powerful mathematical framework across numerous domains, providing principled methods for comparing and transforming probability distributions. machine learning, OT has been applied to generative modeling (Arjovsky et al.) (017), domain adaptation (Courty et al., 2017), and representation learning (Tolstikhin et al. 2018). Beyond traditional applications, OT has become foundational to modern generative modeling paradigms: flow matching (Lipman et al. 2023 uses optimal transport displacement interpolation to define probability paths for training conuus normalizing flows, while reetified flows (Liu et al. 2023) employ OT principles to learn straight transport paths for efficient. generative modeling. In graphics and computer vision, OT enables geometry processing (Solomon et al.) 2015), point cloud analysis (Bonneel et al. 2016), image color transfer (Ferradans et al. 2014 and texture synthesis (Gao et al. 2019).
The growing importance of OT in scientific computing is demonstrated by its application to GSA Wiesel 2022: Borgonovo et al., 2024), where OT-based sensitivity indices offer valuable theoretical properties for understanding complex black-box models. However, classical OT methods face severe computational limitations when applied to real-world problems, motivating the development of neural approaches that can scale to modern machine learning applications.
2.2 Neural Optimal Transport
Neural optimal transport methods were initially developed to address the computational bottlenecks of classical OT solvers. Early approaches include Wasserstein GANs (Arjovsky et al., 2017), which approximate Wasserstein distances through adversarial training but do not compute explicit transport maps. Later methods
leveraged Brenier's theorem (Brenier] 1991 to implement. Monge maps using input convex neural networks
(ICNNs) (Amos et al., 2017 Makkuva al. 2020
However, ICNN-based approaches suffer from fundamental limitations that restrict their applicability. The convexity requirement imposes strict architectural constraints: all weights must be non-negative, activation functions are limited to convex choices (typically RcLU), and spocialized initialization procedures are required to maintain convexity during training. More recently, Korotin et al. (2023) introduced Neural Optimal Transport (NOT) and Kernel NOT [Krotiin et al. 2022), which bypass ICNN limitations through a minimax formulation that supports general cost functions and stochastic transport maps. NOT uses unconstrained neural networks for transport maps and critic functions, enabling flexible architectures while maintaining theoretical guarantees. We build upon this work for conditional OT formulation.
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2.3 Conditional Transport Challenges
Traditional approaches conditional transport problems require solving multiple independent OT problems for different conditioning values. This becomes computationally prohibitive at scale- for instance, global sensitivity analysis applications require computing OT- sensitivity indices by solving hundreds to thousands of individual OT problems for each input variable and partition (Borgonovo) ct al. 2024). This approach is intractable for high-dimensional models with large datasets due to memory requirements for storing cost matrices and the combinatorial explosion of required solvers.
Computational Bottlenocks:
Methodological Limitations: Existing conditional OT methods suffer from architectural constraints that prevent effective scaling. CondOT (Bunne et al., 2022) relies on ICNNs, inheriting all the expressivity and optimization problems discussed above, which are further amplified in conditional settings where different. conditions may require qualitatively different transport behaviors. Alternative approaches (Wang et al.) 2023 impose architectural restrictions and assumptions the impul distributions that limit their flexibilily. Furthermore, these methods typically use simple conditioning strategies like concatenation, which cannot model the different transport behaviors required when conditioning variables significantly alter the underlying distribution structure.
Benchmarking Limitations: Unlike other areas of machine learning, conditional optimal transport lacks standardized benchmarking and evaluation protocols. This absence hinders systematic comparison across different approaches and limits progress in the field. Existing methods often evaluate on different datasets with varying experimental setups, making it difficult to assess their relative strengths and weaknesses. Moreover, existing approaches focus on implementing the neural OT solver, while we also test various conditioning architectures.
2.4 Conditioning Mechanisms in Generative Models
The design of effective conditioning mechanisms is crucial for adaptive generative models. Conditioning approaches have evolved significantly, creating spectrum of expressivity and computational complexity.
Simple concalenation (Ho et al., 2020 Mirza & Osindero 2014) directly combines feature vectors with
conditioning information but lacks the flexibility to model condition-deeeneent behaviors. More sophisticated approaches include classifier guidance (Dhariwal & Nichol, (021). which uses external classifiers to guide generation; normalization- based mothods like FiLM (Perez et al. 2018) and AdaIN (Huang & Belongic 2017)
which modulate feature statisties; and allention mechanisms (Vaswani el al. 2017 Rombach el. al. 2022
which allow selective focus on relevant conditioning information.
At the high end of the expressivity spectrum, hypernetworks (Ha et al. 2017 dynamically generate network parameters based on conditioning inputs, enabling fundamentally different computations for different condi- tions. This capability is particularly valuable for OT applications, where different conditioning values may require distinct mapping strategies that cannot be achieved through feature modulation alone.
For conditional transport maps specifically, the choice of conditioning mechanism is critical. Simple con- catenation, as used in CondOT (Bunne al.) 2022), fails when different conditions require qualitatively different transport behaviors for example, when transporting between fundamentally different distribution types or when conditioning variables induce non-linear changes in the optimal transport structure. Our hypernetwwork-based approach addresses this limitation by generating condition-specific transformation pa- rameters, providing the expressiveness of separate networks per condition while maintaining computational efficiency through parameter sharing in the hypernetwork itself.
This work provides the first systematic comparison of conditioning mechanisms within the neural optimal transport framework, demonstrating the superior performance of hypernetwork based approaches for learning flexible conditional transport maps.
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Stochastic samples (from =)
C C
P
Noise
Condition
c1
Input
T Tαx 2, c)
Conditional Transport Map
T(
T(x, 2g)g)
Figure 1: A diagram of our conditional OT map framework. Our transport network T takes as input samples I r and transports them to Q, conditioned by c. Optionally, T can receive additional noise inputs z ~ S from a known probability distribution 8, introducing stochasticity.
3 Neural Conditional Transport Maps
Our solution extends the Neural Optimal Transport (NOT) framework of Korotin et al. (2023) with a flexible conditioning architecture that enables efficient learning of conditional transport maps across both categorical and continuous conditioning variables. At. the core of our method lies a hypernetwork that dynamically generates transport layer parameters based on conditioning inputs, providing high expressivity while maintaining computational efficiency through shared parameter generation. This design allows different conditions to induce different mapping stralegies essential when conditional distributions require distinct transport behaviors that cannot be captured through simple feature concatenation modulation approaches.
Our framework offers several key advantages over existing methods: (1) it avoids the architectural constraints of ICNN approaches, enabling flexible network designs with standard optimization procedures; (2) it scales efficiently to large numbers of conditioning values without requiring separate OT solvers, addressing the computational bottlenecks in sensitivity analysis applications; (3) it provides unified mechanism for handling both discrete and continuous conditioning variables through learnable embeddings and positional encoding: and (4) it enables systematic comparison of conditioning strategies within the samne neural OT formulation.
We illustrate our conditional OT maps in Figure ! In this section, we begin by formalizing the conditional OT problem (Section 3.1 then describe our architecture that leverages encoder-decoder structures for both transport map T' and critic function f (Section 3.2h. We then introduce our conditioning mechanism that handles discrete variables and continuous partitions (Section 3.3 present architectural variants including our hypernet work approach and alternative conditioning strategies (Section 3.4,, and detail our training procedure with custom pretraining strategy that addresses the optimization challenges common in conditional transport learning (Section 3.5 Implementation details are provided in the supplementary material.
3.1 Problem formulation
To address the computational bottlenccks conditional OT applications, we formulate the problem following the neural OT framework that enables efficient scaling large numbers of conditioning values without requiring separate solvers for each condition. Let us consider two probability measures μ and defined over two Polish spaces, X' C mm and y C R", respectively. Let's also define a lower-semicontimouus cost function k: x - [0, + such that k(y.v') = 0 if and only if n'. The Kantorovich formulation of the OT problem can be stated as follows:
K(H,v) inf xell(..) xxy
k(x, y) dr(
(1)
where 11(,,)) v) is the set of joint probability measures on X y with marginals and p.
Even though the primal formulation of the OT problem in equation easier to interpret, the dual formulation is more useful from practical perspective, especially using neural networks. First, we introduce the concept.
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Combinc
Discrete Cond.
e/
MLP
Cont.
Cond.
PE(p)
PF(p)
e gd
Output
Input
by, ad
M ad
[
Encoder
Conditioning
Decoder
Noise
ResBlocks
c
ResBlocks
Figure 2: Architecture of the conditional networks (transport T and critic f). The encoder processes input data [| for T or for f) through residual blocks to produce latents h, E R. We combine discrete variable embeddings eμ with positional encoding of contimmous values PF(p) through an MLP to produce the unified conditioning vector с Rd., The conditioning module transforms this latent into E Rd which the decoder processes into the final output. The noise z is only used in
of k-transform. The k-transform jk function f:y - R is defined as (*) = inf inf f(u)). The dual formulation of the Kantorovich OT problem is then:
K(J,2) = sup J (x))d(()) - I, flu)dr(u)
(2)
For our purposes, the dual problem can be further reformulated as a maximin problem (Korotin al. 2023 We introduce third atomless distribution, w, defined over the Polish space R.. Given a mcasurable map T:Xx Z + and its push-forward operator T # the maximin formulation is:
K(μ v) = sup inf ((S.T),
(3)
T
where
C(I.T) 1/000/1000 f(y) duls) + / k(a, K(c,,Te,,
/ ()) dp(x)).
(4)
This formulation allows our neural approach amortize computation across conditioning values, avoiding the prohibitive cost of solving separate OT problems for each condition. We note here that the results in Korotin et al. (2023) can be extended to weak costs, but they are out of the scope of this work.
The conditioned problem. We start from equation 4 to integrate the conditioning mechanism into the problem formulation, enabling applications like global sensitivity analysis to compute transport- indices efficienty across multiple conditioning values with a single learned model. We assume that the condition belongs to a measure space C. From the theoretical perspective, the extension straightforward:
K(n,v.)) sup supinf inf C(f,Tcc).
(5)
f
where
[(J,T,c) = / fly, f(u.))del))+ / ((,,T(,, S(T(2,,,)). du(a))
(6)
3.2 Neural Optimal Transport
Building on the NOT framework of Korotin et al. 2023 we parametrize both the transport map T and critic function f using neural networks. We employ encoder-decoder architectures that enable flexible conditioning in the latent space while supporting various layer types (linear, convolutional, attention) based on the data modality. We illustrate our model design in Figure
Transport & Critic: The transport map X' y is implemented as a neural network with encoder- decoder structure. For the common where X Ra the encoder maps Ra into Rd conditioning

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Algorithm 1 Training Neural Conditional Optimal Transport
Input: Distributions P, Q, 3 accessible by samples; Transport network Tg: x RR x + R Critie network fw: RQ x C R; Cost L:X x x + R; Conditioning distribution C Maximum steps N, Transport iterations per step KT
Pre-training: Initialize To and Jw using objectives in Eq. (3)-(7)
3: Output: Learned conditional transport map Tg
4: for t 1,2...., N do
5: Sample conditioning eNC
Sample batches ~ Q P; for each x E X sample Zz ~
Lg + [X] Exex 12,1 Exez, (x, 2,)), [8] 2yer (y(
8: Update w using OC, (gradient ascent)
du
for = 2,...... do
10:
Sample conditioning c~C
11: Sample batch X N~ P;; for cach E X sample 2z ~3
12: LT - + [(z, Te(a, Fu, [z.1 ExE9, fo (T
1
13: Update 0 using
PCr 00
14: end for
15: end =0
module transforms Rd into Rd, and the decoder maps Rd into RR. The complete transport map is: T(z,2, (0,2,,) c) Decodery
z)). where [a, 2] E g=+x denotes the concatenation of input and noise d is the latent dimension, and d' is the conditioned latent dimension. The critic function f y x C + R follows the same structure without noise input: ((y = Decoder DeeooeDetCCCaCaaitiiniinnn sennttiioning(Enndde (w)dddrrrwy...e
Layer Design: Unlike CondOT (Bunne et al. 2022 which uses ICNNs, our unconstrained architecture
supports flexible designs with standard optimizatiin procedures. We use residual blocks He et al. (2016) for both T and f, along with layer normalization Ba al. (2016) and orthogonal initialization Saxe al. (2014). Each block consists of (esBlock(h) h + Fh)), where is learnable scaling parameter and represents composite function that may include normalization, non-linear activations, and appropriate transformations for the data type.
The encoder produces a sequence hidden states hh... where hy E Rd is used for conditioning. The decoder takes the conditioned representation and maps to the output space. Both modules can be instantiated with different layer lypes depending on the data modality: Encoder(z) = hr, where h4 Lhyer,(hg-1) and ho æ. Here, Layer; can be any differentiable layer. The encoder learns condition- invariant features from the source distribution P, while the decoder applics condition-speeific transformations. Importantly, this flexibility allows Layer, to include any slandard neural network components.
Encoder-Decoder Design:
The conditioning operates on the latent representation h2 E gd from the encoders. Given a condition a E C, the conditioning function transforms the latent representation before passing it to the decoder. This design allows the network to learn condition-specffic transformations while sharing feature extraction across conditions.
Latent Space Conditioning:
3.3 Conditioning mechanism
Our framework supports conditioning on discrete and continuous variables, with the capability to enable either both types of conditioning depending the application. The conditioning module transforms these inputs into a unified representation that modulates the transport map, enabling efficient scaling to applications requiring transport across many conditioning values without solving separate OT problems.
For discrete variables (e.g., categorical features with K possible values), we use learnable embeddings. Each variable & (0, 1.... K-
- 1} is mapped to an embodding eş E Rdc, where de is the condition dimensionality, defined as e₁ E(k] with E Kxd. For continuous variables, based on Transformers Vaswani et al. 20117 and Radiance Fields Mildenhall et al. (2020), we use sinusoidal positional encodings to preserve the continuous
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nature while providing rich representation: PE(p,2i) siu( 10006 ). (E(p. (E(p,2i+1) cos 10000 where is
the min-max normalized contimuous value, d is the encoding dimension, and i {0, 1,.,, d/_.,,d/222 -1) Additionally, we support other encoding strategies including Fourier features, learned embeddings, or scalar values, allowing flexible adaptation to different problem domains.
Unified Conditioning. The conditioning module flexibly handles discrete variables, continuous variables, or both. When both types are present, their representations are concatenated, then processed to produce the final conditioning vector MMP ([ediscrate: Peoottiiiuuuus ]],, where [:--] denotes concatenation and MLP is a multi- perceptron with SiLU Ramachandran et al. (0177) activations. The resulting vector E R is
used to modulate the transport map in the latent space.
Flexible Configuration. Both discrete and continuous conditioning are optional and can be independently enabled or disabled. The module requires at least one type of conditioning to be active. This flexibility allows our framework to adapt to various application domains from purely categorical problems to continuous spaliotemporal transport lasks. Unlike CondOT (Bunne et al. 2022), we use different conditioning embeddings for T and f, as this showed better performance. This separation proves to be quantitatively important, suggesting that the transport map and critic function use conditioning information in fundamentally different ways-the transport map must adapt its mapping strategy based on conditions, while the critic must evaluate transport quality conditional on the same information.
3.4 Conditioning modules variants
We explored several designs for applying the conditioning E Rd to modulate the transport map.
Hypernetwork Conditioning The hypernetwork generates the final layer weights and biases of the encoder based on the conditioning. Given the encoder output h E Rd and conditioning c, the hypernetwork H is a shallow MLP that generates parameters: [W. b] = H(c). where W ædd*, ".bbbbb é R". The final output then computed as: y hW + b. Unlike feature modulation approaches that only adjust existing computations, generating weights enables fundamentally different transformations per condition addressing the core limitation of simple conditioning strategies that fail when different conditions require qualitatively distinct transport behaviors. This approach provides significant expressiveness while maintaining the computational efficiency of a single shared architocture, directly addressing the scalability bottlenecks in applications like global sensitivity analysis.
Alternative Conditioning Mechanisms We evaluated several conditioning strategies beyond our hyper- network approach. The simplest baseline is Concatenation, which directly combines feature vectors with the condition but cannot model the fundamenially different transport behaviors required when conditions significantly alter the optimal mapping strategy. More sophisticated approaches include Cross- Atten-itt (Vaswani et al., 2017). which uses attention mechanisms to modulate features, and Feature-wise Linear Modulation (FiLM) (Perez et al. 2018), which applies learnable transformations to existing features. However, these methods are fundamentally limited to adjusting or re-weighting existing computations rather than enabling entirely different computational paths per condition. We also tested normalization- based methods like Adaplice Inslance Normalizalion (Huang & Belongie 2017) and Conditional Layer Normalization (Su et al. 2021), along with attention-inspinn techniques such Squeeze-and Excite (Hu et al.) 2018) and Feature-wise Affine Normalization (FAN) (Zhou et al.] 2021), but these approaches similarly cannot generate the qualitatively different transport behaviors required when conditions induce fundamentally different optimal mappings. Although these alternatives showed promise in specific scenarios, our hypernetwork approach consistently demonstrated superior performance across all benchmarks. Notably, our lightweight hypernetwork implementation proved to be computationally efficient in training time, making a sound choice even for low-budget scenarios. We ablate these components on Section 5
3.5 Training Procedure
During training, the transport map T' and critic function f must maintain adversarial balance while adapting to diverse conditioning values. This creates optimization challenges, where networks must simultaneously
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learn transport mappings and adapt to diverse conditioning values. We address them through a two-phase approach: pre-training for stable initialization followed by minimax optimization.
Pre-training: Randomly initialized networks may implement highly non-linear transformations far from identity, leading to unstable optimization problem that is particularly acute in conditional transport learning where networks must balance transport objectives with condition-dependent adaptation. Motivated by this observation, we introduce lightweight pre-training that establishes favorable initial conditions for both networks. We pre-train T to approximate the identity mapping: DT E.- --cl,--ll/Te 2. = This is important in conditional settings because the transport map must learn to interpolate between potentially very different optimal mappings for different conditioning values- baseline that prevents the network from collapsing to degenerate solutions that work for some conditions but fail for others. This initialization ensures that, early in training, the transport map preserves input structure. Besides, we pre-train f with a multi-objective loss: cpre = Ammoohh A Smmoohh
from identity provides stable
+ Atraspoort/rrauppo + Amag -mag- First, the smoothness term prevents sharp discontimuities that lead to gradient explosion during adversarial training:Cemooth cccllff + e,c) fax c) c)]l2] where ~ .N(00 o²). ) In our formulation, f must evaluate transport quality consistently across diverse conditioning values, making smoothness important to prevent the adversarial training from destabilizing when conditions require very different transport behaviors. Sccond, the transport term maintains the OT objective: Liransport = Coo.P..sS..-cc((((((( c), c)] - Ey~q.c-- e)]. This helps f learn meaningful discrimination between transported and target samples from initialization. Finally, a magnitude control term prevents unbounded growth, enhancing stability: Laug = Ey~q,e-cl (()
Following pre-training, we employ an alternating training that reflects the minimax structure of optimal transport. We detail our approach in Algorithm | Following Korotin et al. (2023), we perform Kr 1 updates for the transport map per critic update. This asymmetry addresses the imbalance in learning complexity-die the transport map must learn condition-dependen transformations while maintaining transport constraints, whereas the critic primarily serves as a measure of transport quality. We find Kr E [4, 6J provides an adequate balance between training stability and efficiency. This asymmetric training schodule proves essential for conditional transport learning, where the complexity imbalance between transport and critic functions is amplified by the need to handle diverse conditioning values.
Optimization:
The performance of this learning procedure depends how we sample from the conditioning space C during training. For discrete variables, we use uniform sampling across all categories. For continuous variables, we employ Beta distribution sampling Beta(o, 3) with symmetric parameters (a = 3 = 0.95), which slightly oversamples values around the min and max values in the training datasets. This strategy prevents underfitting in boundary conditions. This sampling strategy is important for applications like global sensitivity analysis, where transport quality must be maintained across the full range of conditioning values to ensure accurate sensitivity indices.
Conditions Sampling:
4 Applications and Experimental Setup
We evaluate our conditional neural transport framework across diverse applications that span scientific com- puting, sensitivity analysis, and image generation. These applications are specifically chosen to demonstrate the breadth of conditioning scenarios our framework can handle: from purely discrete conditioning to hybrid discrete-continuous conditioning, and from lower-dimensionel scientific problems to image generation tasks. Each application presents unique challenges that test different aspects of our conditional transport approach, providing comprehensive validation of the framework's 's capabilities.
4.1 Climate Economic Impact Distribution Transport
We examine the economic impacts of climate change using the empirical damage function model of Burke et al. (015)). This application requires building an emulator for complex multivariate distributions conditioned on both categorical (scenario) and continuous (time) variables, representing a challenging test case for hybrid conditioning mechanisms.
Problem Formulation.
Let C R represent the space of GDP per capita with climate damages across 20 countries. For each country i, the impact is quantified through 1000 bootstrap replicates, resulting in
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high-dimensional empirical distribution that captures the uncertainty in damage estimates. We define conditioning space C = Csap x Cyear, where Csep {0, 2, represents four SSP scenarios (SSP1-1.9, SSP2-4.5, SSP3-7.0, 5SP5-8.5) and Cyear = [2030, 2100] represents projection years. Our goal is to learn a conditional transport map Ty : R" x Z x C - X' that efficiently transforms samples from a reference distribution to match larget distributions under different climate scenarios and future years.
Dataset and Evaluation. The dataset contains empirical damage distributions for 20 countries across SSP scenarios and 71 years (2030-2100), totaling 5,680 conditioning combinations. Each distribution is characterized by 1000 bootstrap samples, providing rich uncertainty quantification. We use 80% of the data for training and 20% for evaluation, ensuring temporal and scenario stratification. We evaluate performance using the Wasserstein-2 distance between generated and target distributions-
4.2 Global Sensitivity Analysis for Integrated Assessment Models
Our second application focuses global sensitivity analysis (GSA) for the RICE50+ Integrated Assessment Model (IAM)
Gazzotti 2022), using the optimal transport- baasport--beed sensitivity indices presented in Borgonovo et al. (024). This application demonstrates our framework's s ability to replace computationally intensive traditional approaches with efficient neural approximations.
Thcoretical Background. Let Y (Yi,..., Y2) represent quantities of interest as a function of inputs C= (O1, (O1,..,,Ca) Assume C and Y random vectors on probability space (, with Pc, and bY denoting the distributions of C, and Y, respectively. Consider model f: C Rd + R The OT-based sensitivily index for the i-th input is:
[o,1K (Y,C)) E|K(Y, YO]
where K(py, Pyic;) is the OT cost between unconditional and conditional output distributions, and the denominator provides normalization. This index satisfies zero-independence (c (Y, C;) 0 iff Y independent of C;) and max-functionality (1 (Y, = iff Y g(C;) for some function g).
Problem Formulation. Let f CCCR + c 58 be the RICE50+ model mapping input paramcters C E C to output Y Y y. RICE50+ Gazzotti 2022) is an IAM with high regional heterogeneity used to assess climate policy benefits and costs. We estimate the sensitivity of CO2 emissions for a single region to three inputs related to emissions abatement costs (klogistic), aversion to inter-country inequality (gamma), and climate impacts (kw_2), leveraging data from Chiani et al. (2025).
Traditional computation of these indices requires solving multiple OT problems for each conditioning variable and partition-sseeificalll specifically, for d variables with M partitions each, this requires x M separate OT solutions, becoming computationally intractable for large-scalc applications. We partition each input space into M = 25 bins and define conditioning space č {0, 1,2) [0, 1], where the first component represents variable index and the second represents normalized partition. Our approach learns a single conditional transport map Ty : y'x 2 C y that can compute sensitivity indicos for all variables and partitions efficiently.
Dataset and Evaluation. The dataset contains 10,000 model evaluations with corresponding input-oulput pairs. We use the traditional estimator with partitioned inputs as ground truth, computed using transportation simplex solvers We evaluate our model with the Pearson correlation between neural and simplex-computed costs.
4.3 Image Generation Tasks
To demonstrate the generality of our approach beyond scientific computing, we evaluate conditional image generation on standard computer vision benchmarks. These tasks test our framework's ability to handle high-dimensional visual data with both discrete class labels and continuous attribute conditioning.
MNIST Conditional Generation. We perform conditional digit generation with discrete conditioning, using class labels C {o,...,91 9). For this application, we use convolutional models for both T and f, which
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include convolutional layers with different values for strides and kernel sizes, therefore leveraging standard components for image generation models without imposing convexity constraints.
Problem Formulation. We learn a transport map Ta 2 x C + 282x28 that transforms noise into digit
images conditioned on class and attributes.
Datasct and Evaluation. We use the standard MNIST dataset augmented with rotation and stroke thickness variations create ground truth conditional distributions. To evaluate this setup, we use the Fréchet Inception Distance (FID) for generation quality, and digit classification accuracy.
4.4 Experimental Protocols
Hardware and Implementation. All experiments are conducted on RTX 4070 GPU with 17-13700H CPU. Our framework is implemented in PyTorrh with automatic mixed precision training. We use CodeCarbon Lacoste et al. 2019 to monitor energy consumption across experiments.
Architecture Details. For climate and GSA applications, we use fully connected encoder-decoder ar- chitectures with encoder layers (hidden size 128) and decoder layers for transport networks, and 3 encoder /ecoder layers for critic networks. For image generation, we employ convolutional architectures with residual blocks. All networks use layer normalization, orthogonal initialization, and learnable residual scaling parameters
Training Configuration. We employ our two-phase training procedure: 500 steps of pre-training followed by alternating minimax optimization. Transport networks receive Kr 5 updates per critic update. Learning rates are x 10-5 for transport and 3 x 10 for critic networks, with AdamW optimization and weight decay of 0.03. Conditioning sampling uses uniform distribution for discrete variables and Beta(0.95, 0.95) for continuous variables.
Experimental Comparisons. Our evaluation focuses systematic ablation studies to identify optimal design choices within our framework. We compare different conditioning mechanisms including concalenation, cross-attention, Feature-wise Linear Modulation (FiLM), normalization- based approaches (Adaptive Instance Normalization, Conditional Layer Normalization), and attention-inspired techniques (Squeeze-and-Exxiee Feature-wise Affine Normalization). We also evaluate different continuous variable encoding strategies (positional encoding, Fourier features, scalar values), architectural choices (shared vs. separate conditioning embeddings). and training procedures (with and without pretraining). For GSA applications, we validate against traditional transportation simplex solvers to establish ground truth sensitivity indices.
Comparisons with CondOT Bunne et al. 2022 We additionally compare against two CondOT baselines. CondOT defoult uses the original CondOT architecture and optimization recipe, together with a tokenized factorized conditioning scheme in which categorical and discretized continuous condition components are represented as separate conditioning tokens. Since the original CondOT formulation is not designed for mixed catogorical: continuous conditioning in the form considered here, continuous variables (time indices partition levels) must. be discretized before being provided to the model. To provide a more informative comparison, also define CondOT adapted variant, in which replace this discretized conditioning with mixed conditioning mechanism combining one-hot categorical inputs and normalized continuous scalars. We do not include CondOT in the MNIST experiments, since its original formulation is not designed for convolutional architectures. For both configurations, we create architectures with a comparable number of trainable parameters to our models, to fully isolate the impact of architecture design and optimization choices rather than model capacity. Number of training epochs remain equal to our models, while optimization choices (optimizer betas, number of inner iterations) for CondOT follow those of their original implementation.
5 Results and Analysis
We present a comprehensive empirical evaluation of our conditional neural transport framework across the applications described in Section g Our evaluation follows a systematic approach: we first establish the effectiveness of our architectural improvements through ablation studies both unconditional and conditional
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Baseline
+Orth.
+Nomm
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Metric
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Res
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Black Hole Wass.
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82.15
100
100
100
Figure 3: Results of our ablation study comparing different model configurations on an unconditional transport setting. The table presents quantitative metrics, while the right panel shows the visualization.
settings, then demonstrate the practical impact of our approach real-world applications. Finally, analyze computational aspects and discuss the implications of our findings.
5.1
Ablation Studies
To validate our design choices and quantify the contribution each component, we conduct systematic ablation studies examining architectural improvements and conditional-speciti design decisions.
5.1.1 Architectural Foundation
We first evaluate architectural improvements to the base NOT framework Korotin et al. 20023 establishing a strong foundation for our conditional extensions. Building upon their ReLU-based architectures, we incrementally add orthogonal initialization, residual connections, and layer normalization, while preserving identical training configuration and comparable parameter counts across models.
Figure 3 presents results across three datasets that test different aspects of neural optimal transport. The first two show 2D probability distributions using simple MLPs as baseline architecture. We measure KL divergence and Wasserstein loss between target and transported distributions (from 2D Uniform prior), using 2 samples. The third row showcases convolutional model performing image-to-Image generation on MNIST (digit> to digit 3), evaluated with MMD distance, Wasserstein loss, FID score, and classification accuracy.
In both MLP and convolutional settings, our cnhancements yield notable quantitative and qualitative improvemenss, as shown in the visualization pancls. These architectural enhancements form the foundation of our conditional transport experiments, demonstrating that improved convergence be achieved without. increasing computational cost.
5.1.2 Conditional Framework Design
Having established our architectural foundation, we turn to the key design decisions specific to conditional transport learning. Table I presents our systematic evaluation conditioning mechanisms, encoding strategies, and training procodures.
We compare our final configuration using hypernetwork conditioning, pretraining, positional encoding, and separate conditioning embeddings for transport and critic networks- with systematic variations. Training times and parameter counts provide computational cost analysis, while application-specffe metrics evaluate transport quality. For the climate damages datasel, we report Wasserstein distance and efficiency (Wasserstein x time). For LAM data, we report Pearson correlation between neural and simplex-computed computed costs and afficinccy-adjnstdd correlation. All experiments use identical training protocols on RTX 4070 GPU with 17-13700H CPU, measuring 0.009323 kWh electricity consumption via CodeCarbon for the climate damages application Lacoste al. 2009
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Table 1: Ablation study our conditional transport framework. We report computational cost and accuracy on our datasets. Our final model (leftmost column) uses hypernetwork with positional encoding and pretraining, without shared embedding. Each other column group represents variations from this configuration. We present best results for our method in green bold, second best are underlined, worst are red. Note that for the MNIST experiment, there is no continuous input variable, therefore don't ablate their encodings. CondOT default uses the original CondOT architecture and optimization recipe with tokenized factorized conditioning over categorical and discretized continuous variables. CondOT adapted keeps the CondOT training recipe unchanged but replaces this discretized conditioning with mixed conditioning combining one-hot categorical inpuls and normalized continuous scalars. CondOT evaluated only on the climate and GSA/RICE tasks; MNIST entries are marked as not applicable.
CondOT
Pretrain
Embed.
Conditioning Type
Continuous Eucoding
Ours
Defanit
Adapted
False
Shared
Concnt
SE
FiLM
FAN
Asta.
CLN
AdaIN
Sealer
Fourier
Comp. coat
Time 4 586
721
157
354
851
487
921
712
658
464
Parsmeterr (k) 4
1216
1316
1158
1158
1156
1324
2764
2874
2861
2908
2791
2790
1150
11522
Climate Damages
Wass.
0.322
0.255
0.265
0.267
0.716
0.974
0.458
0.721
0.691
0.744
0.256
0.172
Wass. x time
188.7
189.8
79.22
96.51
121.1
120.1
396.7
8318
223.0
6610
191.9
289.6
118.3
83.59
IAM Data
0.025
0.360
0.942
0.905
0.812
0.305
0.577
0.456
0.388
0.441
0.612
0.682
0.894
0.901
p/thmo(x100)
0.004
0.064
0.202
0.200
0.177
0.069
0.068
0.053
0.079
0.048
0.096
0.104
0.193
0.192
MNIST
Time (6)
N.A.
314
306
314
277
293
319
314
326
315
311
N.A.
Parameterr (m))
N.A.
2.893
2,893
2,803
1.815
1.855
1.961
1.832
1.866
1.832
1.965
N.A.
Accuracy
N.A.
1.000
1.000
1.000
0.992
0.600
1.000
1.000
0.500
1.000
1.000
N.A.
FID
N.A.
2.509
2,658
3.167
3.565
11.18
2.904
2. 732
5.792
2.855
2.912
N.A.
FID
N.A.
3245
956.1
867.8
1868
899.3
905.6
N.A.
Several important patterns emerge from these results. The conditioning mechanism choice significantly impacts both training efficiency and accuracy. Our lightweight hypernetwork consistently outperforms alternative like feature modulation (FiLM) or adaptive normalization approaches (AdaIN, CLN) without major increases in training cost or parameters. Complex approaches like cross-attention prove suboptimal in both accuracy and computational cost. While concatenation offers lowest training time, it achieves worst accuracy on the IAM dataset, confirming our hypothesis that simple conditioning fails when conditions require fundamentally different transport behaviors. Interestingly, adaptive normalization mechanisms like FAN, AdaIn CLN are good alternatives to the hypernetwork in image generation tasks, which is perhaps unsurprising given their success in different computer vision problems. This also suggests that these alternatives would scale consistently better than cross-attention mochanisms, achieving both relatively expressive transformations while having a reduced training cost.
The continuous variable encoding strategy also proves crucial across both datasets. Positional encoding (our approach) and Fourier features achieve the best accuracy, though Fourier features are less computationally efficient. This holds across both applications despite different continuous variable semantics, suggesting that appropriate processing of raw conditioning information plays vital role in expressivity.
Our analysis of embedding sharing strategics shows that separating conditioning embeddings for transport and critic networks leads to accuracy gains. This suggests that transport and critic functions utilize conditioning information in fundamentally different ways. This has important implications for conditional transport architecture design. Finally, our pretraining procedure enables higher accuracy with minimal additional training cost, demonstrating that appropriate data-driven initialization improves training dynamics and final results without efficiency penalties.
Comparisons with previous work When comparing our framework against the CondOT baselines, the results in [ show improvements both expressivity and computational efficiency. On the climate damages datasct, our method outperforms both CondOT Default and CondOT Adapted. In the GSA application, CondOT Default fails to capture the underlying sensitivity structures (p = 0.025), and while the Adapted version improves this (p = 0.389), both fall short of our hypernetwork approach (p 0.942). These results
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SSP1-2.6
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Figure 4: Results of our climate damages model, under different SSP scenarios, particular country the
dataset. We show the ground truth distribution of damages and samples from our model.
show that both adequate conditioning mechanisms designs and removing the requirements of partial convexity from neural OT architectures allow for more expressive models without increasing overall computational cost.
5.2
Application Results
Building on the insights from our ablation studies, we now demonstrate the practical effectiveness of our approach across the scientific and computational applications described in Section
5.2.1 Climate Economic Impact Distributions
Figure 4 presents our model's performance on the climate damages dataset, demonstrating its capability to generate realistic distributions across different climate scenarios. For each SSP scenario, compare ground truth GDP per capita with climate damages distributions against samples from our conditional transport. model over the 2030-2100 time horizon.
The results demonstrate our model's effectiveness at capturing both central tendencies and uncertainty (shaded regions represent 90% confidence intervals) specific each scenario. Our approach successfully learns the distinct temporal evolution patterns across different SSPs: SSP1 shows relatively stable relative impacts with wide uncertainty reflecting the optimistic but uncertain nature of this scenario, while SSP2 exhibits moderate decline after 2060 consistent with intermediate climate projections. The SSP3 scenario shows the most pronounced downward trond-represcnting severe climate impacts-which which our model captures accurately despite the smaller uncertainty bounds characteristic of this more deterministic high-impact scenario.
5.2.2 Global Sensitivity Analysis
Figure 5 presents our validation against traditional simplex-based approaches for computing OT-based sensitivity indices across three input variables in the RICE50+ model. This comparison directly tests our framework's ability replace computationally intensive traditional methods with efficient neural approximations.
The first three panels show transport costs across different partition values (0 to 1) for each variable, with simplex results in blue and our neural method in red. Our neural transport approach closely tracks the simplex cost patterns across all three variables, effectively capturing the complex sensitivity structure of the underlying model. For the klogistic variable (emissions abatement costs), both methods identify similar regions of high scnsitivity, with our neural approach maintaining strong correlation (p = 0.942 from Table D with simplex results. The gamma variable (inequality aversion) shows more complex sensitivity patterns that method accurately reproduces, including the pronounced peak partition value 0.25. For ku
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gutama
kw 2
Partition Valuae
Partition Value
Partition Valuuo
Figure 5: Comparison of simplex and our neural transport across three variables for the 1AM dataset. The first three panels show costs across partition values for klogistic, gamma, and kw variables (simplex in blue, neural in red). The rightmost panel shows average costs for each variable.
(climate impacts), both methods detect lower overall sensitivity with consistent patterns across partition values.
The rightmost panel summarizes average transport costs for each variable, revealing that our neural method preserves the relative importance ordering among variables while maintaining comparable a absolute cost. maguitudes. This preservation of variable ranking is critical for GSA applications where accurate identification of the most. influential parameters drives decision- deciiioonmakkng in policy and scientific contexts.
Crucially, our neural approach achieves this accuracy with dramatically improved ocalability-requiine single trained model rather than solving hundreds of individual OT problems (75 separate optimizations for this variablar,a 25-partition case). This computational advanlage becomes increasingly important model complexity and conditioning dimensionality grow, enabling sensitivity analysis for problems previously considcred computationally intractable.
5.2.3 Image Generation
Figure e presents qualitative results of our ablation study, on MNIST image generation task, using our Conditional Optimal Transport framework. As shown, our baseline configuration (top row) achieves clean digits, significantly outperforming other conditioning methods. Removing the pre-training step or sharing the conditioning between T and f yield small but noticeable artifacts in the generated digits. Note that MNIST generation is a relatively trivial task for image generative models, and these networks can be trained in approximately minutes on a standard laptop. How these results would scale up to largo-scale image generation is outside the scope of this work, as Neural Optimal Transport is not state-of saaee-offthh-rttt formulation for high-resolution rollabee image generation, where diffusion models or large-scale generative adversarial networks are currently among the best-performing alternatives.
5.3 Computational Analysis and Discussion
The experimental results demonstrate that our conditional neural transport framework successfully addresses the key limitations identified in existing approaches while providing practical advantages for real-world applications.
Our ablation studies reveal that the hypernetwork conditioning mechanism provides the best balance of expressivity and computational efficiency. Unlike feature modulation approaches that can only adjust existing computations, hypernetworks generate fundamentally different transformation parameters for each condition-osscntia when optimal transport maps must adapt qualitatively across conditioning values. The computational overhead of our lightweight hypernetwork implementation proves minimal compared to the expressivity gains achieved.
The importance appropriate continuous variable encoding, demonstrated consistently across both climate and GSA applications, highlights an often-overlooked aspect of conditional transport learning. Our sinusoidal ionalional encoding effectively captures multi-scale variation in continuous conditioning variables, enabling
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Figure 6: Qualitative ablation study MNIST digit generation, comparing method with hypernetwork (top row) with ablated components, including the lack of pretraining (second row), the used of shared conditioning for T' and f (third), and the different conditioning types we tested.
smooth interpolation between conditioning values while maintaining sufficient expressiveness for complex
conditional relationships.
The success of our pretraining strategy confirms the importance of initialization in conditional transport learning. By starting transport networks near identity mappings and stabilixing critic functions through multi-objective pretraining, we avoid the optimization pathologies that plague conditional transport training, particularly when handling diverse conditioning values that may require very different optimal mappings.
For practical applications, our approach enables new possibilities in scientific computing and uncertainty quantification. The climate modeling results demonstrate effective emulation of complex multivariate distributions under hybrid conditioning. At the same time, the GSA application shows how computational bottlenecks in sensitivity analysis can be overcome through efficient neural approximation. The preserved statistical properties and correlation with traditional methods indicate that our neural approach can be a reliable replacement for classical techniques in appropriate contexts.
Future work should explore extensions to higher-dimension conditioning spaces and investigate theoretical guarantoes for conditional transport learning. The framework's modular design makes it well-suited for adaptation emerging applications in generative modeling, domain adaptation, and scientific computing, where conditional optimal transport principles provide natural problem formulations.
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6 Conclusions
We presented a neural framework for conditional optimal transport that addresses fundamental limitations in existing approaches. Our hypernetwork- based conditioning mechanism addresses three critical bottlenecks: the computational explosion of solving hundreds of individual OT problems in applications like global sensitivity analysis, the architectural constraints of -asse methods that limit expressivity, and the inability of simple conditioning strategies to expressively model transport behaviors across conditions. Our hypernetwork architecture generates transport layer parameters dynamically based on conditioning inputs. This provides the expressiveness of separate networks per condition while maintaining computational efficiency- Our design enables different mappings for different conditions, essential when conditional distributions require distinct optimal transport strategies that feature modulation approaches cannot capture. Importantly, we achieve this wout introducing constraints on the neural network architectures (we demonstrate rate successful cases of MLPs and CNNs), achicving high generality and scope in terms of model design and application domain.
Experiments across scientific computing and image generation tasks show superior performance compared alternative conditioning approaches. Systematic ablation studies confirm the value of each architectural component. For global scnsitivity analysis, framework replaces hundreds of individual OT problems with a single trained model, achieving high correlation (o 0.942) with traditional methods while enabling previously intractable analyses. For climate economic modeling, we efficiently emulate complex multivariate distributions under hybrid conditioning with preserved statistical properties.
Limitations and Future Work. Our evaluation focuses primarily on leedforward architectures with residual connections. Recurrent or attention-based architectures could potentially improve performance for sequential conditioning scenarios. While we handle discrete, continuous, and hybrid conditioning effectively, we have not explored complex modalities like CLIP embeddings or pixel-wise semantic labels that could enable more sophisticated generative modeling.
Our hypernetwork approach, though efficient relative to separate networks per condition, still costs more computationally than simple concatenation. More efficient hypernetwork designs or adaptive conditioning mechanisms could reduce this overhead. Theoretical analysis of convergence properties and approximation guarantces would strengthen the foundation for critical applications. Future directions include multi-modal conditioning, connections lo diffusion models and normalizing flows, and applications to dynamical systems and causal inference. The modular framework design makes it well-suited for adaptation to emerging generative modeling paradigms and scientific computing applications.
Broader Impacts. Our work addresses computat ional barriers in scientific computing and uncertainly quantification. Making conditional optimal transport computationally tractable enables more comprehensive sonsitivity analysis for complex models in climate science, cconomics, and policy decisions and scientific understanding. The approach also democratizes access advanced optimal transport techniques for researchers with limited computational resources. For generative modeling, our framework enables more sophisticated conditional generation with principled transport-based approaches, complementing existing diffusion and flow-based methods. This could accelerate progress in controlled content generation, data augmentation, and domain adaptation. However, the ability to generate realistic samples conditioned on specific attributes could be misused for creating mislcading synthetic media or other harmful content. Appropriate safeguards include access controls for sensitive applications, combining automated analysis with domain expertise for critical decisions, and providing educational resources to help practitioners understand limitations. The substantial benefits for scientific computing and uncertainty quantification outweigh potential risks when proper oversight is implemented.
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Acknowledgments
Carlos Rodriguez-Parde Leonardo Chiani, and Massimo Tavoni acknowledge support from the European Research Council, FRC grant agreement number 101044703 (FUNICE) CUP D87G2200340006.
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Reviews:
[{"review_id": "zuhpqZgb5n", "reviewer": "Reviewer_5UEZ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper extends the Neural Optimal Transport (NOT) framework to the conditional setting, enabling transport maps that depend on both categorical and continuous auxiliary variables. The authors propose a hypernetwork-based conditioning mechanism that generates condition-specific parameters for the transport model, allowing different conditions to induce qualitatively distinct mappings without requiring separate OT solves.\n### Key strengths:\n- Provides a practical approach for amortizing conditional OT computation, reducing the need for repeated OT solves across many conditions.\n- Hypernetwork conditioning offers greater expressiveness than simple concatenation or modulation, and the empirical study systematically compares multiple conditioning architectures.\n- Demonstrates applicability to real-world scientific modeling tasks (climate, IAMs) and OT-based sensitivity analysis, where conditional transports are computationally prohibitive with classical OT solvers.\n### Key weaknesses:\n- The conceptual contribution is incremental, as conditional neural OT was previously introduced (e.g., Supervised Training of Conditional Monge Maps, NeurIPS 2022); the main novelty lies in adopting hypernetworks for conditioning.\n- The architectural change, which is generating transport-layer weights via an extra hypernetwork, may be viewed as a modest variation rather than a substantial methodological advance.\n- The paper could more clearly articulate the theoretical advantage of hypernetwork conditioning over existing conditional OT parameterizations.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The paper shows strong experimental results:\n- Figure 4 – Climate Damage Model: Shows conditional transport under a continuous scenario parameter\n- Figure 5 – IAM Data: Shows conditional transport across categorical scenario partitions, comparing multiple conditioning mechanisms, with the proposed hypernetwork performing best.\n- Table 1 – Ablation Study: Summarizes results across tasks, showing that hypernetwork conditioning with pretraining consistently improve performance over alternatives.", "audience_yn": "Yes", "audience_explanation": "Researchers working on optimal transport, conditional generative modeling, and scientific applications such as climate or sensitivity analysis may find the empirical insights and practical conditioning strategies of interest, even if the methodological novelty is limited.", "requested_changes": "- Provide a visualization or toy example illustrating the claimed limitation of feature-modulation conditioning (e.g., FiLM, concatenation) and how hypernetwork-generated weights enable 'fundamentally different transformations' across conditions. A simple 2D synthetic example where conditions require qualitatively different transport patterns would help substantiate this claim.\n- Add discussion comparing the proposed approach to prior neural OT methods, especially conditional neural OT models (e.g., Supervised Training of Conditional Monge Maps, NeurIPS 2022). Please explain specifically:\n  - What forms of conditioning these prior methods used\n  - Why those mechanisms are insufficient for the scenarios considered here\n  - How hypernetwork-based weight generation overcomes those limitations in practice", "broader_impact_concerns": "No major ethical or societal concerns are apparent. The work focuses on methodological advances in optimal transport and conditional modeling, and does not involve sensitive data, human subjects, or deployment-related scenarios.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "jmDqpjPfgM", "reviewer": "Reviewer_23Ab", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper introduces a novel conditioning method for Neural Optimal Transport maps, based on a new hypernetwork architecture that generates parameters for optimal transport maps dynamically based on conditioning inputs. The authors test their approach on various datasets, including climate modeling, global sensitivity analysis, and image generation tasks. The paper is well written and provide an extensive introduction and reference to prior work.", "claims_and_evidence_yn": "No", "claims_explanation": "The paper extends the Neural Optimal Transport framework (Korotin 2023) with a conditioning module based on on hypernetwork. While the architectural design and algorithms are well explained, the experimental results mainly show ablation studies. \n- In section 5.1.1 the authors perform an ablation study of architectural and initialization hyperparameter choices from Korotin et al. 2023. It is unclear why such an analysis contributes to the experimental validation of the paper's claims about conditional NOT.\n- In section 5.1.2 an extensive ablation study is performed on 3 datasets, extensively evaluating how each architecture and initialization choice impacts the evaluation. However, providing at least one orthogonal method of learning conditional OT maps, such as CondOT would be useful to provide an additional reference for model performance, which does not consist of variations of the NOT framework for Korotin et al. 2023", "audience_yn": "Yes", "audience_explanation": "Yes, since it provides at least one novel approach to conditional NOT, leveraging hypernetworks, as well as a comprehensive ablation study on three datasets for other approaches to feature-modulation-type conditioning.", "requested_changes": "- The paper primarily compares against CondOT (Bunne et al., 2022), although only in scope, since it does not report any back to back comparison with ConOT. \n- The paper lacks comparison with the more recent work by Wang et al. (2025) that they cite. Given that Wang et al. (2025) also address conditional OT with neural networks, a direct empirical comparison would strengthen the paper's claims. In Wang et al. various approaches are presented. I would suggest the author to extend section 5.1.2 with at least one orthogonal approach to the conditional NOT introduced here, in order to provide a meaningful performance reference to other conditional OT approaches (as the ones presented in Wang et al. 2025)\n- The paper claims efficiency advantages but only tests on small-scale problems (20 countries, 58-dimensional outputs, MNIST). Would it be possible for the authors to at least compare a few ablations/baselines with datasets like CIFAR-10 ?\n- The paper would benefit from a more concise and clear description of the architecture. The key novelty of the paper, the use of a hypernetwork, is briefly described in section 3.4, first paragraph, but for example, it is not clear how the hypernetwork is trained, and where it fits in the algorithm. 1. Furthermore, a schematic example of how the hypernetwork conditioning compares to standard conditioning (essentially what is explained in section 3.4) would be beneficial and highly increase clarity.\n- The choice of Beta(0.95, 0.95) for continuous sampling needs better justification.\n- In algorithm 1, the notation L(x, T_θ(x,Z̃_x, c)) is unclear—should this be the cost function k?", "broader_impact_concerns": "None", "additional_comments": "The hypernetwork approach is well-motivated and the experimental validation spans interesting applications. However, the paper would benefit from stronger baselines and more extensive scalability analysis.", "extra_scores": {}, "extra_text": {}}, {"review_id": "NFfWdtHdP3", "reviewer": "Reviewer_yeP3", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "Authors introduce neural conditional transport maps: a neural network that transports samples from the source distribution to the target distribution. The transport map is conditioned on an auxiliary variable and optionally stochastic. More specifically, the final approach utilizes hypernetwork that generates the weights of the final layer of the encoder based on the unified conditioning vector.\n\nStrengths: \n* The paper is clearly written. I am not very familiar with optimal transport literature but the paper provides a clear introduction to the field that enables understanding of author's contributions.\n* The experimental part has sufficient ablation studies and the method is evaluated on 3 different application.\n\nWeaknesses: \n* The datasets used in applications seem very toy.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The paper provided strong theoretical foundations. The experimental part contains thorough ablation study regarding the architecture of encoder and decoder networks as well as design of conditioning. The method is evaluated in multiple settings, however, the used datasets seem toy.", "audience_yn": "Yes", "audience_explanation": "I believe this work would be relevant for audience interested in optimal transport.", "requested_changes": "None of the following changes is critical but they would strengthen the work:\n* Figure 2 provides a general schema for conditional networks. It would be nice to see the diagram of the final approach that utilizes hypernetworks as conditioning. Final approach does not fully match Figure 2 where conditioning is applied on the output of the encoder but hypernetwork generates the weights of the final layer of that encoder which was confusing for me at the beginning.\n* Table 1: it is unclear what experiment times and parameters refer to.\n\nMinor writing errors/typos: \n* Section 4.1 ends with \"-\"\n* Figure 4: \"on a particular country the dataset\"", "broader_impact_concerns": "Broader Impacts section is sufficient for a paper investigating the fundamentals.", "additional_comments": "I am not very familiar with optimal transport literature. Therefore, my confidence is quite low.", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)