Multi-Task Learning of Conditional Mean Operators: applications to dynamical systems and uncertainty quantification

Sami Chemlal; Thibaut Germain; Rémi Flamary; Vladimir R. Kostic; Karim Lounici

stat.MLSep 28, 2026

Multi-Task Learning of Conditional Mean Operators: applications to dynamical systems and uncertainty quantification

Authors: Sami Chemlal, Thibaut Germain, Rémi Flamary, Vladimir R. Kostic, Karim Lounici

Organizations: CMAP, Ecole Polytechnique · University of Novi Sad

Abstract

Estimating conditional statistics and learning representations of a population of conditional distributions are central problems in many data-driven applications, including uncertainty quantification and dynamical systems analysis. Conditional mean operators (CMOs), a class of linear operators between function spaces, resolve these objectives by providing access to a broad class of conditional statistics. However, existing methods typically estimate each CMO independently or constrain it to prespecified function spaces, thereby preventing the exploitation of shared structure across related distributions. In this work, we posit that related CMOs share finite-dimensional input and output function spaces, and are specialized for each task with a linear operator mapping these spaces. Based on this hypothesis, we introduce MTL-CMO, a multi-task framework that jointly learns shared function spaces and task-specific operators across multiple datasets. We further introduce T-CMO, a transfer learning method that reuses the shared spaces to estimate, in closed form, the operator of a new conditional distribution. We establish statistical guarantees quantifying the benefits of jointly learning the shared function spaces. Our experiments demonstrate that learning shared function spaces improves uncertainty quantification across a broad range of conditional distributions and, when applied to Langevin and plasma dynamics, yields compact representations of complex dynamics that retain physically meaningful information and enable parameter identification.

Figures & tables

	Method	CD1	CD2	CD3	CD4
Multi-task	MTL-CMO (ours)	0.088 $\pm$ 0.004	0.102 $\pm$ 0.002	0.106 $\pm$ 0.003	0.070 $\pm$ 0.001
	Pooled-NCP ( Kostic et al., 2024 )	0.091 $\pm$ 0.002	0.120 $\pm$ 0.009	0.139 $\pm$ 0.008	0.077 $\pm$ 0.008
	MTL-MDN ( Bishop, 1994 )	0.110 $\pm$ 0.001	0.138 $\pm$ 0.005	0.216 $\pm$ 0.009	0.107 $\pm$ 0.005
	DeepJMQR ( Rodrigues and Pereira, 2020 )	0.091 $\pm$ 0.002	0.114 $\pm$ 0.005	0.125 $\pm$ 0.008	0.066 $\pm$ 0.004
Single-task	NCP ( Kostic et al., 2024 )	0.096 $\pm$ 0.001	0.167 $\pm$ 0.037	0.185 $\pm$ 0.001	0.150 $\pm$ 0.002
	CFM ( Lipman et al., 2023 )	0.112 $\pm$ 0.001	0.182 $\pm$ 0.021	0.170 $\pm$ 0.006	0.150 $\pm$ 0.005

Table 1: Mean 1-Wasserstein distance to the ground-truth conditional distribution averaged over 100 tasks. Scores in

<\!\text{mean}\!>\pm<\!\text{std}\!>

over 10 training seeds. Lower is better. First and second .

Figure 1: Comparison between MTL-CMO and the single-task NCP for representing CDFs on four conditional distribution families. Performances measured with the 1-Wasserstein to the ground truth.

Figure 2: Transfer learning evaluation on unseen conditional distributions. Comparison in

1

-Wasserstein between T-CMO and the single-task NCP depending on the number of observations.

	Trajectory length $N$
Method	5 000	10 000	20 000	40 000	80 000
T-CMO (ours)	$\mathbf{0.261\pm 0.148}$	$\mathbf{0.190\pm 0.108}$	$\mathbf{0.140\pm 0.075}$	$\mathbf{0.099\pm 0.055}$	$\mathbf{0.075\pm 0.039}$
Single-task NCP ( Kostic et al., 2024 )	$0.297\pm 0.138$	$0.224\pm 0.098$	$0.176\pm 0.068$	$0.138\pm 0.047$	$0.115\pm 0.031$
Pooled NCP ( Kostic et al., 2024 )	$0.306\pm 0.143$	$0.219\pm 0.102$	$0.163\pm 0.078$	$0.118\pm 0.058$	$\underline{0.087\pm 0.041}$
RFF/EDMD ( Li et al., 2017 )	$0.295\pm 0.140$	$0.233\pm 0.123$	$0.177\pm 0.069$	$0.135\pm 0.045$	$0.109\pm 0.030$
RFF/Laplace ( Kostic et al., 2025 )	$0.296\pm 0.141$	$0.221\pm 0.100$	$0.172\pm 0.072$	$0.132\pm 0.051$	$0.106\pm 0.034$

Table 2: Reconstruction error between estimated and ground truth operators. Hilbert-Schmidt error in

<\!\text{mean}\!>\pm<\!\text{std}\!>

over the

150

unseen systems. Lower is better. First and second .

Figure 3: Recovery of $3^{rd}$ eigenfunction of an unseen Müller-Brown system with T-CMO/Laplace and its single-task counterpart NCP/Laplace. Global error in cosine distance.

Figure 4: Reconstruction error across eigenfunctions and potential values between T-CMO/Laplace and NCP/Laplace. Error averaged over the

150

unseen systems. Global error in cosine distance.

Figure 5: Left: t-SNE on the distance matrix between the $320$ training operator spectra colored by $g$ and $\kappa$ . Right: parameter identification with kernel regression evaluated on the $80$ held-out systems.

Figure 6: Identification of the parameters

g

and

\kappa

versus trajectory length for T-CMO and competing methods.

Appendix figures & tables25 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Representative conditional densities from CD1 , CD3 , and CD4 for different conditioning inputs and task parameters.

	Hidden layers	$d$	$r_{k}$	Activation	Dropout
CD1	$3\times 64$	$64$	$20$	GELU	$0.15$
CD2	$2\times 64$	$64$	$7$	Tanh	$0.2$
CD3	$4\times 64$	$64$	$18$	GELU	$0.14$
CD4	$4\times 64$	$64$	$13$	Tanh	$0.16$

Appendix

Table 3: Neural architecture and operator rank used for the four conditional-distribution families.

	Epochs	Batch size	$\mathrm{lr}_{\mathrm{shared}}$	$\mathrm{lr}_{\mathrm{specific}}$	$\mathrm{wd}_{\mathrm{shared}}$	$\mathrm{wd}_{\mathrm{specific}}$	Grad. clip	Scheduler
CD1	$5400$	$32$	$5.8\times 10^{-5}$	$1.4\times 10^{-4}$	$2.2\times 10^{-3}$	$2.6\times 10^{-2}$	$5.0$	None
CD2	$4200$	$64$	$2.7\times 10^{-4}$	$6.4\times 10^{-5}$	$3.9\times 10^{-3}$	$2.6\times 10^{-4}$	$2.0$	None
CD3	$3600$	$64$	$1.2\times 10^{-4}$	$1.8\times 10^{-3}$	$2.2\times 10^{-4}$	$3.4\times 10^{-6}$	$1.0$	None
CD4	$2600$	$64$	$3.0\times 10^{-3}$	$2.9\times 10^{-3}$	$9.1\times 10^{-6}$	$4.4\times 10^{-3}$	$2.0$	Cosine

Appendix

Table 4: Optimization parameters for the uncertainty-quantification experiments.

$n_{\mathrm{tr}}$	50	100	150	200	250	300	350	400
CD1
NCP	0.2362 $\pm$ 0.0059	0.1739 $\pm$ 0.0025	0.1455 $\pm$ 0.0038	0.1258 $\pm$ 0.0021	0.1152 $\pm$ 0.0015	0.1062 $\pm$ 0.0021	0.1004 $\pm$ 0.0018	0.0948 $\pm$ 0.0024
T-CMO	0.2629 $\pm$ 0.0053	0.1869 $\pm$ 0.0015	0.1546 $\pm$ 0.0042	0.1315 $\pm$ 0.0012	0.1188 $\pm$ 0.0020	0.1072 $\pm$ 0.0022	0.1004 $\pm$ 0.0015	0.0943 $\pm$ 0.0015
NCP / T-CMO	0.90	0.93	0.94	0.96	0.97	0.99	1.00	1.01
CD2
NCP	0.2326 $\pm$ 0.0082	0.1826 $\pm$ 0.0041	0.1704 $\pm$ 0.0016	0.1588 $\pm$ 0.0015	0.1538 $\pm$ 0.0024	0.1500 $\pm$ 0.0007	0.1470 $\pm$ 0.0024	0.1458 $\pm$ 0.0028

Appendix

Table 5: Transfer to unseen conditional distributions. Wasserstein-

1

distance to the ground-truth conditional distribution (mean

\pm

standard deviation over

10

target-data seeds), averaged over

100

unseen tasks. Lower is better. The last row of each block reports the ratio between single-task NCP and T-CMO.

Figure 8: Sensitivity of transfer performance on CD4 . Left: Wasserstein-

1

error as a function of the number of observations per source task

n

, with

K=200

. Right: Wasserstein-

1

error as a function of the number of source tasks

K

, with

n=200

Varying $n$ (samples per task), $K=200$
	T-CMO (ours)	NCP	Gain
$n=50$	0.281 $\pm$ 0.015	0.283 $\pm$ 0.013	1%
$n=100$	0.169 $\pm$ 0.005	0.229 $\pm$ 0.007	26%
$n=200$	0.099 $\pm$ 0.003	0.186 $\pm$ 0.005	47%
$n=500$	0.068 $\pm$ 0.002	0.143 $\pm$ 0.005	52%
$n=1000$	0.058 $\pm$ 0.001	0.128 $\pm$ 0.003	55%

Appendix

Table 6: Sensitivity of transfer performance on CD4 . Wasserstein-

1

distance to the ground-truth conditional distribution. Gain denotes the relative reduction in

W_{1}

of T-CMO with respect to single-task NCP. Lower is better.

Figure 9: Parametric Müller–Brown family. Top: varying $r$ at fixed $\theta=0^{\circ}$ modifies the relative depth of the third metastable basin while largely preserving its orientation. Bottom: varying $\theta$ at fixed $r=0.95$ rotates the local geometry of the same basin over a $60^{\circ}$ range.

Figure 10: Spectral-rank diagnostic. Mean reduced-rank reconstruction error of the whitened lagged operator as a function of the retained rank. Most of the reduction occurs within the first three components, after which the error rapidly saturates.

Figure 11: Geometry of the learned Müller–Brown operators. t-SNE of pairwise S-GOT distances between operators learned during the multi-task stage, colored by $r$ (left) and $\theta$ (right).

Figure 12: Multi-task versus single-task CMO learning. Eigenfunction errors of MTL-CMO versus independently trained single-task CMO models. Points below the diagonal favor MTL-CMO.

Architecture
Configuration	Value
Input dimension	$2$
Shared feature dimension $d$	$64$
Hidden layers	$3\times 576$
Activation	SiLU
Task-specific rank	$20$

Appendix

Table 7: Architecture, training, and spectral-estimation configuration for the Müller–Brown experiment.

$\psi_{1}$
	Trajectory length $N$
Method	2 000	5 000	10 000	20 000	40 000	80 000
T-CMO (ours)	$\mathbf{0.0731\pm 0.0966}$	$\mathbf{0.0294\pm 0.0414}$	$\mathbf{0.0151\pm 0.0206}$	$\mathbf{0.0079\pm 0.0095}$	$\mathbf{0.0040\pm 0.0053}$	$\mathbf{0.0021\pm 0.0026}$
RFF-EDMD	$0.0982\pm 0.1537$	$0.0333\pm 0.0427$	$0.0248\pm 0.0821$	$0.0098\pm 0.0098$	$0.0058\pm 0.0054$	$0.0037\pm 0.0026$
MetaKoopman	$0.0837\pm 0.1154$	$0.0327\pm 0.0424$	$0.0174\pm 0.0209$	$0.0099\pm 0.0098$	$0.0058\pm 0.0055$	$0.0040\pm 0.0029$
Single-task NCP	$0.0839\pm 0.1071$	$0.0332\pm 0.0415$	$0.0177\pm 0.0206$	$0.0101\pm 0.0095$	$0.0059\pm 0.0054$	$0.0041\pm 0.0026$

Appendix

Table 8: Eigenfunction errors for the first three non-trivial modes on the

150

unseen Müller–Brown systems. Values are mean

\pm

standard deviation. Lower is better; best and second best per column.

$\lambda_{1}$
	Trajectory length $N$
Method	2 000	5 000	10 000	20 000	40 000	80 000
T-CMO (ours)	$\mathbf{0.8288\pm 2.8035}$	$\mathbf{0.1919\pm 0.2668}$	$\mathbf{0.1241\pm 0.1057}$	$\mathbf{0.0916\pm 0.0824}$	$\mathbf{0.0643\pm 0.0569}$	$\mathbf{0.0443\pm 0.0367}$
RFF-EDMD	$0.9852\pm 2.8284$	$0.3160\pm 0.3197$	$0.2710\pm 0.1682$	$0.1236\pm 0.1025$	$0.2239\pm 0.0935$	$0.1620\pm 0.0706$
MetaKoopman	$1.0705\pm 3.0177$	$0.2779\pm 0.4465$	$0.1842\pm 0.1614$	$0.1356\pm 0.1260$	$0.1043\pm 0.0853$	$0.0870\pm 0.0689$
Single-task NCP	$1.0360\pm 2.9714$	$0.2714\pm 0.4275$	$0.1829\pm 0.1489$	$0.1346\pm 0.1168$	$0.1026\pm 0.0793$	$0.0885\pm 0.0590$

Appendix

Table 9: Relative eigenvalue errors for the first three non-trivial modes on the

150

unseen Müller–Brown systems. Values are mean

\pm

standard deviation. Lower is better; best and second best per column.

Figure 13: Transfer versus single-task spectral recovery. (a) Distribution of mode-wise

1-\cos_{\pi}(\widehat{\psi}_{j},\psi_{j})

errors over unseen systems. (b–d) Error distribution across potential-energy levels for

\psi_{1}

\psi_{2}

, and

\psi_{3}

Figure 14: Spatial recovery of the first three eigenfunctions on a representative unseen system. From left to right: finite-difference reference, T-CMO, single-task NCP, and the corresponding pointwise errors. Eigenfunctions are sign-aligned and normalized in $L^{2}(\pi)$ .

Figure 15: Representative electrostatic potential $\phi$ (top) and vorticity $\Omega=\Delta_{\perp}\phi$ (bottom) across the $(g,\kappa)$ domain. Each field is standardized independently for visualization.

Architecture
Configuration	Value
Input size	$3\times 64\times 64$
Shared feature dimension $d$	$128$
Task-specific rank	$80$
Encoder	Convolutional ResNet
Residual stages	$(64,128,256)$

Appendix

Table 10: Architecture and optimization configuration for the plasma experiment.

Figure 16: Evolution of the MTL-CMO training objective over

120

epochs.

Figure 17: Mean one-step RRR reconstruction error across the

320

source systems versus retained rank. The transfer rank is fixed to

32

Figure 18: MDS embeddings of the dynamical representations of the $320$ source systems, colored by $g$ (top) and $\kappa$ (bottom).

Figure 19: Physical-parameter recovery on held-out systems versus trajectory length for T-CMO and competing representations.

Interchange parameter $g$
Trajectory length $T$	$256$	$474$	$878$	$1\,625$	$3\,008$	$4\,093$
T-CMO (ours)	$\mathbf{0.225}$	$\mathbf{0.516}$	$\mathbf{0.778}$	$\mathbf{0.878}$	$\mathbf{0.897}$	$\mathbf{0.932}$
MIDST	$0.178$	$0.326$	$0.421$	$0.564$	$0.761$	$0.763$
Pooled NCP	$0.172$	$0.339$	$0.438$	$0.546$	$0.587$	$0.700$
RFF (128)	$-0.096$	$0.156$	$0.217$	$0.511$	$0.669$	$0.692$
Density-gradient parameter $\kappa$

Appendix

Table 11: Held-out

R^{2}

for physical-parameter recovery versus available trajectory length. Lower-dimensional spectral methods use

62

features; MIDST uses its native

128

-dimensional representation.

Figure 20: Held-out

R^{2}

versus the number of slow generator modes retained from the T-CMO representation. Each mode contributes two spectral coordinates.

Features $D$	$32$	$64$	$\mathbf{128}$	$256$	$512$	$1024$
Interchange $g$	$0.388$	$0.662$	$\mathbf{0.714}$	$0.641$	$0.639$	$0.639$
Density gradient $\kappa$	$0.968$	$0.979$	$\mathbf{0.986}$	$0.985$	$0.983$	$0.982$

Appendix

Table 12: Gaussian RFF sensitivity to the random-feature dimension

D

on complete trajectories.

Parameter	Value
Source systems	$320$
Held-out systems	$80$
T-CMO / Pooled NCP spectral features	$62$
RFF dictionary dimension	$128$
RFF spectral features	$62$
MIDST features	$128$

Appendix

Table 13: Physical-identification evaluation settings.

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Department of Mathematics, University of California Los Angeles, Los Angeles, CA 90095, USA.

Probabilistic Operator Learning Neural Operators Multi--Task Learning

Open paper AskPDF arXiv

We study the approximation and statistical complexity of learning collections of operators in a shared multi-task setting, with a focus on the Multiple Neural Operators (MNO) architecture. For broad classes of Lipschitz multiple operator maps, we derive near-optimal upper bounds for approximation and statistical generalization. On the lower-bound side, we establish a curse of parametric complexity and prove corresponding minimax rates. Together, these results show that shared representations across tasks do not increase the overall cost: multi-task operator learning follows the same scaling laws as single operator learning. We also compare MNO with a multi-task extension of DeepONet based on concatenated task inputs and show that, from a worst-case approximation-complexity perspective, both architectures satisfy essentially the same asymptotic rates.

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