Physics-Constrained Fine-Tuning of Flow-Matching Models for Generation and Inverse Problems
Authors: Jan Tauberschmidt
Sophie Fellenz
Sebastian Josef Vollmer
Andrew B. Duncan
Venue: ICLR 2026 Poster (2026)
Keywords: Generative Modeling
Physics‑Informed Machine Learning
Inverse Problems
Parameter Identification
Primary area: generative models
TL;DR: We fine-tune pretrained flow-matching models using weak-form PDE residual rewards to generate physically consistent fields and infer latent parameters for inverse problems without paired solution–parameter training data.

Abstract: We present a framework for fine-tuning flow-matching generative models to enforce physical constraints and solve inverse problems in scientific systems. Starting from a model trained on low-fidelity or observational data, we apply a differentiable post-training procedure that minimizes weak-form residuals of governing partial differential equations (PDEs), promoting physical consistency and adherence to boundary conditions without distorting the underlying learned distribution. To infer unknown physical inputs, such as source terms, material parameters, or boundary data, we augment the generative process with a learnable latent parameter predictor and propose a joint optimization strategy. The resulting model produces physically valid field solutions alongside plausible estimates of hidden parameters, effectively addressing ill-posed inverse problems in a data-driven yet physics-aware manner. We validate our method on canonical PDE problems, demonstrating improved satisfaction of physical constraints and accurate recovery of latent coefficients. Further, we confirm cross-domain utility through fine-tuning of natural-image models. Our approach bridges generative modelling and scientific inference, opening new avenues for simulation-augmented discovery and data-efficient modelling of physical systems.

Published as a conference paper at ICLR 2026
PHYSICS-CONSTRAINED FINE-TUNING OF FLOW- MATCHING MODELS FOR GENERATION AND INVERSE PROBLEMS
Jan Tauberschmidt Sophie Fellenz", Schastian J. Vollmer1.2, Andrew B. Duncan
DSA, German Research Center for Artificial Intelligence (DFKI)
Department of Computer Science, University of Kaiserslautern-Lnndaa (RPTU)
3 DDepartment of Mathematics, Imperial College London {jan. tauberschniddt sebastian, vollmer (tl.mmerreakkk.kk de, fellenzeca acc. uni-ki de, a.
ABSTRACT
We present framework for fine-tuning flow- Now-matching generative models en- force physical constraints and solve inverse problems in scientific systems. Start- ing from model trained on low-fidelity or observational data, we apply a dif- ferentiable post-training procedure that minimizes weak-form residuals of gov- erning partial differential equations (PDEs), promoting physical consistency and adherence to boundary conditions without distorting the underlying learned dis- tribution. To infer unknown physical inputs, such as source terms, material pa- rameters, boundary data, we augment the generative process with learnable latent parameter predictor and propose a joint optimization strategy. The result- ing model produces physically valid field solutions alongside plausible estimates of hidden parameters, effectively addressing ill-posed inverse problems in a data- driven yet physics-aware manner. We validate our method on canonical PDE prob- lems, demonstrating improved satisfaction of physical constraints and accurate recovery of latent coefficients. Further, we confirm cross-domain utility through fine-tuning of natural-image models. Our approach bridges generative modeling and scientific inference, opening new avenues for simulation-augmentt discov- ery and data-efficient modeling of physical systems.
INTRODUCTION
Physical systems with rich spatio-temporal structure can effectively represented by deep genera- tive models, including diffusion and Now-matching methods (Kerrigan et al.) 2024 Erichson et al.) 2025 Baldan et al.] 2025) Price et al.) (2023). Although their dynamics can be highly complex, these systems often governed by fundamental principles, such conservation laws, symmetries, and boundary conditions, that constrain the space of admissible solutions. Incorporating such physical structure into generative modeling can improve both sample fidelity and out-of-distribttinn general-
ization.
In many scientific domains, including atmospheric and oceanographic modeling, seismic inversion, and medical imaging, we often observe system states without access to the underlying physical parameters that govern them. Crucially, PDE-based constraints are typically purameter-depender with residuals that vary according to material properties, source terms, or other latent variables. Prior work has largely focused on simple or global constraints- such as fixed boundaries or symmetries, that apply uniformly across the data distribution. Handling derpedetr-dependden constraints naively would require training over the joint distribution solutions and parameters, which often infea- sible because parametric labels are missing, expensive obtain, high-dimensional. Addressing this limitation is critical for scientific discovery. Many inverse problems in the natural sciences and engineering require reasoning about unobserved parameters exploring hypothetical scenarios inaccessible to direct experimentation. A generative model that can enforce parameter-dependeee PDE constraints using only observational data would provide powerful tool for data-efficient sim-

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ulation, hypothesis testing, and the discovery of new physical phenomena, helping to bridge the gap between raw observations and mechanistic understanding.
This work proposes framework for fine-tuning flow- Now-matching generative models to enforce parameter-dependent PDE constraints without requiring joint parameter-solution training data. This work aligns with growing trend of simulation-augman
machine learning (Karniadakis et al., 2021). where generative models accelerate scientific discovery by efficiently exploring physically plausible solution spaces. Our approach reformulates fine-tuning as stochastic optimal control problem via Adjoint Matching (Domingo-Enrich et al.] 2025). guided by weak-form PDE resid- uals. By augmenting the model with latent parameter evolution, we enable joint generation of physically consistent solution-parament parameter pairs, addressing ill-posed inverse problems. We evaluate our proposed fine-tuning framework on four representative PDE families spanning elliptic diffusion, elasticity, wave propagation, and incompressible flow and show an application to natural images. We demonstrate denoising and conditional generation capabilities, including robustness to noisy data and the ability to infer latent parameters from sparse observations. Visual and quantitative sults, including strong reductions in residuals across tasks and robustness to model misspecification, highlight the flexibility of our method for integrating physical constraints into generative modeling.
To sum up. our contributions are as follows:
POST-TRAINING ENFORCEMENT OF PHYSICAL CONSTRAINTS: We introduce a fine- tuning strategy that tilts the generative distribution toward PDE-consistent samples using weak-form residuals, improving physical validity while preserving diversity.
FINE-TUNING WITH THEORETICAL GROUNDING: Leveraging the adjoint-matching framework, recast seward-based fine-tuning as stochastic control problem, extending flow-matching models to generate latent parameters alongside states, enabling inverse problem inference without paired training data.
BRIDGING GENERATIVE MODELING AND PIYSSICS-IFORRMD LEARNING: Our ap- proach connects preference-aligned generation with physics-based inference, enabling simulation-auggentte models to generate solutions that respect complex physical laws.
An implementation of our method is available at https:// com/
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RELATED WORK
Constas-Constrained Generative Models Integrating physical constraints- such as boundary conditions, symmetry invariances, and partial differential equation (PDE) constraints-into ma- chine learning models improves both accuracy and offdiittiiutton generalization. Classical ap- proaches, such Physics-Inffrmed Neural Networks (PINNs, Raissi al. (201)), directly regress solutions that satisfy governing equations. While effective for forward or inverse problems, PINNs do not capture distributions over solutions, making them unsuitable for generative tasks that require sampling diverse plausible outcomes.
In the generative setting, the main challenge is ensuring that the physically constrained samples tain the variability of the underlying generative model, avoiding pathological issues such as mode collapse. [Battek al. (2024) proposes unified framework for introducing physical constraints into Denoising Diffusion Probabilistic Models (DDPMs, Ho et al. 22200) at pre-training time, by adding a first-principles ressical loss the diffusion training objective. This loss penalizes viola- tions of governing PDEs (e.g. fluid dynamics equations) so that generated samples inherently satisfy physical laws. The method was empirically shown to reduce residual errors for individual samples significantly, while simultaneously acting as regularizer against overfitting, thereby improving gen- cralization. To evaluate the physics- residual loss, one needs to compute the expected PDE residual of the final denoised sample conditioned on the current noisy state in the DDPM process. Accurately estimating this expectation requires generating multiple reverse diffussii trajectories from the same noisy sample, which makes pre- pra-train significantly more expensive. A common alternative is to use Tweedie's 's formula to approximate the conditional expectation a single pass, but this shortcut introduces bias, particularly in the final denoising steps.
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2uuce
Flow State
#11
denoising, time
S
Flow Purameter
boso
base
Og
a
Figure I: Visual depiction of proposed method. Starting at state 2uuse or z/t, we use the base vector field base to predict the final sample [a]]. Through the inverse predictor y,, we recover the corre- sponding predicted parameters a and á [6]]. These estimates can be used as target for evolving at [c)] or as baseline for the fine-tuned evolution of of [d)]. For purposes of regularization, further consider Ut pointing from the current a, to the predicted final parameter of the base evo-
reg
lution à base [e)].
Zhang & Zou 2025) proposes enforcing constraints through a post-hoc distillation stage, where a deterministic student model is trained from vanilla diffusion model to generate samples in one- step, regularized by a PDE residual loss. In Wang et al.] (2025) the authors introduce PhyDA, diffusion-based data assimilation framework that ensures reconstructions obey PDE-based dynam- ies, specifically for atmospheric science. An autoencoder is used to encode sparse observations into structured latent prior for the diffusion model, which trained with an additional physical residual loss.
Inference- and Cost-Training Constraint Enforcement Various works have proposed ap- proaches to enforce PDE constraints inference time, often in combination with observational constraints, drawing connections to conditional diffusion models (Dhariwal & Nichol 2021 Ho & Salimans 2021). [Huang et al.] (024) introduce guidance terms within the denoising update of score-based diffusion model to steer the denoising process towards solutions which are both consis- tent with data and underlying PDEs. A related approach was considered by Xu al. 22225). further introducing an adaptive constraint to mitigate instabilities in early diffusion steps. In Christopher al.) (2024). the authors recast the inference -etrencce-tmeeeaee sampling of a diffusion process as a constrained optimization problem, each diffusion step is projected to satisfy user-defined constraints or physical principles. This allows strict enforcement constraints, well ODE-based physical laws) on the generated data. [uu & Xu (2024) consider the setting where the base diffusion model is trained on cheap, low-fidelity simulations, leveraging similar approach to generate down-scaled samples via projection.
hard constraints (including convex and non-convex
Flow-Matching Models for Simulation and Inverse Problems Flow-matching (FM, Lipman
al. (2023)) has emerged as a flexible generative modeling paradigm for complex physical systems across science, including molecular systems (Hassan et al. 2024), weather (Price et al.) 2023) and geology (Zhang et al. 2025)- In the context of physics-constrainet generative models Utkarsh et al. (2025) introduces zero-shot inference framework to enforce hard physical constraints in pre-trained flow models, by repeatedly projecting the generative flow at sampling time. Similarly, Cheng et al. (2024) proposed the ECI algorithm, to adapt pre-trained flow-matching model that it exactly satisfies constraints without using analytical gradients. In each iteration of flow sampling, ECI performs: Extrapolation step (advancing along the learned flow). Correction step (applying a constraint-enforremee operation), and an Interpolation step (adjusting back towards the model's trajectory). While projection approaches compelling strategy for hard constraints, they be
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challenging particularly for local constraints such as boundary conditions, as direct enforcement can introduce discontinuities. The above approach mitigates this by interleaving projections with flow steps, however this relies on the flow's ability to rapidly correct such non-physical artifacts.
Baldan et al. (0255) propose Physics-Basee Flow Matching (PBFM), which embeds constraints (PDE or symmetries) directly into the FM loss during training. The approach leverages temporal unrolling to refine noise-free final state predictions and jointly minimizes generative and physics- based losses without manual hyperparameter tuning of their tradeoff. To mitigate conflicts between physical constraints and the data loss, they employ the ConFIG (Liu et al. 2024), which combines the gradients of both losses way that ensures that gradient updates always minimize both losses simultaneously.
Related our approach are the works on generative models for Bayesian inverse problems (Stuart 2010). where the goal is to infer distributions over latent PDE parameters given partial or noisy observations. Conditional diffusion and now-mat models can be used to generate samples from conditional distributions and posterior distributions, supporting amortized inference and uncertainty quantification (Song et al. 2021 Utkarsh al. 2025 Zhang et al., (023). Conditioning is typically achieved either through explicit parameter inputs or guidance mechanisms during sampling, in classifier-g classifier-guided diffusion. While effective when large volumes of paired training data is available, these approaches are less relevant to observational settings where parameters are unobserved. In contrast, our approach connects the observed data to the latent parameters only during post-training, requiring substantially smaller volumes of data.
METHOD
FM models are trained to learn and sample from given distribution of data Pdata. They approximate this distribution by constructing Markovian transformation from noise to data, such that the time marginals of this transformation match those of a reference flow X, B,X, + 7N Specifically FM models learn a vector field v, (z) that transports noise to data, via the ODE dX, = vy (X,, dt. We can optionally inject noise schedule (t) along the trajectory to define equivalent SDE that preserves the same time marginals (Maoutsa et al. 2020).
at))
3e Be
))
dXx, ex(X))++
(x()))
xxx)
X,
dt + (t) be(X,) o() dB, (1)
2m)
where we combine coefficients Be and Y/ into the = =1 ( - -t)-
Assuming we have access to a FM model which generates samples according to distribution p(z), we seek to adjust this model so as to generate samples from the tilted distribution (z) ε e eAr(e)p(z)) where r is a reward function and À characterizes the degree of distribution shift induced by fine- tuning. To achieve this, we leverage the adjoint-matching framework of Somingo-Ennichh et al. (025).. This work reformulates reward fine-tuning for flow-based generative models as control problem in which the base generative process given by hase is steered toward high-reward sam- ples via modifying the learned vector field, which we denote as of with corresponding drift term Bf Our approach is conceptually related reward- or mefference-bsset fine-tuning of generative models (Christiano et al., 2217]; [un et al.) 2024)), where a learned or computed reward steers gener- ation toward desired properties. Here, the reward is defined via PDE residuals, encoding knowledge about underlying dynamics and physical constraints to the solutions space as deviations to differ- ential operators or boundary conditions. Notably, we assume that the distribution generated by the base model p(x) only captures an observed quantity, but does not provide us with corresponding parameters or coefficient fields often needed evaluate the respective differential operator. In the following, we will present a strategy of jointly recovering unknown parameters and fine-tuning the generation process.
3.1 REWARD
A generative model can reproduce the visual characteristics of empirical data while ignoring the physics that governs it, thereby rendering the samples unusable for downstream scientific tasks. To bridge this gap impose the known governing equations soft constraints, expressed through
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differential operators La 0 with parameters a. Throughout, a generated sample is interpreted as a discretization of continuous field æ(E) on a domain Q2 The strong PDE residual is defined as Rannngg (a. a) = f[ 1,200))
In practice, strong residuals involve high-order derivatives that make the optimization landscape unstable. We therefore adopt weak-form residuals of the form (La 15) (*()) for suitably chosen test functions V, which are numerically more stable under noisy misspecified data. Repeated applications of integration by-parts can transfer derivatives from x to v. The set y is composed of compactly supported local polynomial kernels. For each evaluation we draw Nist such functions; their centers and length-scales are sampled at random. A mollifier enforces vllan = 0, justifying the integration by parts. The resulting residual is
Nes
Reweak (I,, a)
Niest Σ (fat, (50))))) L2(D)]
These randomly sampled local test functions act as stochastic probes of PDE violations, providing low-variance, data-efficient learning signal. A more detailed description of the test functions used can be found in Appendix D.3 Note that the residual might be augmented by adding soft constraints for boundary conditions.
3.2 JOINT EVOLUTION
Fine-tuning is nontrivial in our setting because we must infer latent physical parameters jointly with the generated solutions. On fully denoised samples, we can train an inverse predictor, i.e., (((1)) = a1, such that the weak PDE residual is minimized. As naïve approach, this already induces a joint distribution over (f1,a) via the push-forward through yp However, we advocate more principled formulation that evolves both and a along vector fields, enabling joint sampling of parameters and solutions, as well as a controlled regularization of fine-tuning through the Adjoint Matching framework as outlined below. In the fine-tuning model, this can be achieved by directly learning the vector field of.a jointly with of by augmenting the neural architecture. Since no ground-truth flow a for the base model is available, at each state (I,, ai) we define surrogate base flow using the inverse predictor yp. Specifically,
consider the one-step estimates
1 It + (1 t) v, base (xx).
âg =(1)).
The direction from the current state ar the predicted final parameter â serves as a base vector field which we use to evolve a, i.e. utta = (á - (1) () inducing corresponding drift thae This surrogate base flow, starting at noise sample base AO Z). emulates denoising process of the recovered parameter. We denote by a base the parameter aligned with the base trajectory p.hase While the evolution of a does not influence the trajectory of g. the inferred vector field can be used to effectively regularize the generation of the fine-tuned model. Similarly, to regularize towards the parameter recovered under the base model, we introduce additional field "t, a (a.) = (ābase - -at)/(1 !). This vector field points from the current parameter estimate of the fine-tuned trajectory of to the recovered parameter under the base model hase The field is used to pull the fine-tuned dynamics towards final samples associated with parameters similar to those of the base trajectory. The introduced vector fields are visualized in Fig.
3.3 ADJOINT MATCHING
Considering an augmented state variable of the joint evolution X, = -XX;, af)", we cast fine-tuning as stochastic optimal control problem:
min E L 2 11 (x,) f(x.)) dt + s(X1))
s.t. dXx = Ease (X,) + (t) in (Xx,) dt + o(t) dB.
with control (x,)), running state cost f(x)), and terminal cost 9(x,). In this formulation, fine- tuning amounts to wisnt-wise modification of the base drift through application of control ū. i.c.
(X.) Toss (X,) + a (1) (X.).
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In Domingo- Enrochh-rcccheh al. (2025), Adjoint Matching is introduced as technique with lower variance and computational cost than standard adjoint methods. The method is based on a Lean Adjoint state, which is initialized as
of AV & (X2) (Ax v DeVVV((X1,00 ((),, Va Va
and evolves backward in time according to
d az (V (V B (X.)² ā Vxf((x)²)=
J J
Gt.z ar,a
oz
(3)
dt
J'
where the block-Jacobian is evaluated along the base drift for X and a, which means that Jij = vj Li,t0 (Xe, ar) for i,j E {z,a). The hyperparameters A and da can be used to regu- late the extent to which the fine-tuned distribution departs from the base distribution. The Adjoint Matching objective can then be formulated as consistency loss:
X) = 1/2 /// (x,) + (t)) @.
(4)
= 1 %//² (Ilusa(X,,,)) (n) + o() ar 1| + (X,, + a(1) al.al")) dr.
It can be shown (Domingo-Enrich al. 2025) that with f f= O, this objective is consistent with the tilted target distribution for reward -g. if optimized with a memoryless noise schedule. This schedule ensures sufficient mixing during generation such that the final sample X1 independent of Xo- To stabilize fine-tuning we introduce scaled variant of the memoryless noise schedule. Instead of using the canonical choice o² (t) = 21) identified by Domingo-Eni Enrich et (025). we adopt
o² (t) = (1 x)) 217
0 K < 1,
which retains the theoretical memoryless property (see Lemma in Appendix D.4) while attenuat- ing the magnitude of the noise variance. The introduction of the scaling factor 0 < K < constitutes framework. Whereas prior work highlighted
a simple but novel extension of the siijoittmatcchin unique schedule, analysis shows that family of scaled schedules remains consistent with the memoryless condition. This additional degree of freedom acts as a numerical stabilisation knob, mitigating blow-ups near 0 without losing theoretical consistency. Further, it offers a control- fidelity trade-off by regulating the amount of exploration. In practice, this flexibility allows practi- tioners to adapt fine-tuning to the conditioning of the PDE residuals and the stability of the solver, feature not available in the original formulation.
Equation is optimized by iteratively sampling trajectories with the fine-tuned model while follow- ing memoryless noise schedule, numerically computing the lean adjoint states by solving the ODE in Equation and taking a gradient descent step to minimize the loss Equation Note that gradi- ents are only computed through the control üe and not through the adjoint, reducing the optimization target to simple regression loss. We state the full training algorithm and implementation details in Appendix D.5
Adjoint Matching steers the generator toward the reward-tileer distribution, thereby reshaping the entire output distribution rather than correcting individual trajectories. However, when fine-tuning observational data or under system misspecification, we might be interested in retaining sample- specific detail. Empirically we find that this be effectively encoded by imposing similarity of the inferred coefficients between base and fine-tuned model. Therefore, we add a running state cost
f(a) = Xy (a) Tor (a))
which penalizes deviations of the fine-tuned a-drift from the direction pointing toward the base estimate &hass, The hyper-parameter X5 controls smooth trade-off: X, 0 recovers pure Adjoint Matching, while larger Xy progressively anchors the final parameters a obtained under the fine- tuned model to their base-model counterparts, thus retaining trajectory-level detail.
4 EXPERIMENTS
We evaluate across five settings: four PDE systems (including boundary and system misspecifi- cation, and observational noise) and natural-image model. Unlike latent-space fine-tuning for

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images, PDE models operate directly in pixel space. High-variance noise during sampling can drive off-manifold trajectories and perturb PDE residuals, motivating x > for these models. For base Flow Matching backbones, we use U-FNO (Wen et al. 2022 for PDEs and the DiT-based la- tent FM of Dao et al. 2023) for images. In all experiments we first sample from the base generator and pre-train the inverse predictor yp recover a by minimizing the (PDE) residual, then fine- tune. Following Domingo-Enrich et al. (2025), fine-tuning is initialized from the base weights. We augment capacity to condition v/ on at and add separate head for vff Fine-tuning uses mem- oryless noise schedule, while all reported results are generated without injected noise (a(1) 0).
Implementation details appear in App. D.2
Comparisons, ablations, and metries. Our proposed method converts single-variable flow into a joint generative model (Sec. b.2p. We compare against: (i) a Base AM variant (vanilla Adjoint Matching) where yp frozen and used only to compute residuals, (ii) a Base AM+i variant where continues to train but the flow over a is not modeled jointly, and (iii) PBFM (Baldan et al.) (2225), augmented with our pre-trained yp to enable residual evaluation. Details on the comparison methods can be found in App. [.2] All evaluations use 256 samples, generated from shared seeds across methods. We report weak and strong residuals, Rweak and Rstrong. scaled by the mean residual of fixed reference set. The reference set Diet is a synthetic, clean dataset generated under the tar- get PDE specification assumed during fine-tuning (no noise, modified BCs, lossless Helmholtz, or unforced Stokes respectively). We also report Maximum Mean Discrepancy (MMD) based distribu- tional similarities for states and parameters (MMD,, MMD,) computed against this dataset (details While the main text shows representative results, the complete set of experimental evaluations is provided in App.
in App. E.T
4.1 DARCY FLOW
Consider square domain Ω [0, 1]2 where a permeability a<{) and forcing f({) induce pressure field x={) governed by - (a:()) - f({) = 0 with zero Dirichlet boundary conditions and constant f. We draw a from a discretized Gaussian process and corrupt pressures with observation noise before training the base FM. Dataset details are in App. B Figureecompares three Darcy sam- ples generated from the same noise seed Tip the base draw, fine-tuning with our regularization (here =y=1.0), and fine-tuning without regularization. The base pressure hae is visibly contaminated by high-frequency noise, and the inverse predictor &p correspondingly yields scattered, artifact-ridden permeability map a base With regularization enabled, fine-tuning attenuates noise in the pressure x while remaining close to a base Because a base is itself fragmented, some artifacts persist. In contrast, disabling regularization produces fully denoised pressure and markedly more coherent att but at the expected expense of erasing sample-specific details present in the base realization.
Base
Fine-tuned regularization
Fine-tuned w/o regularization
xbase
basg
x²
aa
x²
R
Figure 2: Darcy denoising (qualitative). Base vs. fine-tuned outputs for fixed seed. Regularization
(Ay = 1.0) denoises while staying close to the base sample. Removing it denoises more aggressively at the cost of fidelity to the base realization. Additional non-curated samples in App. F33.I Color maps throughout this work taken from Crameri et al. (2020).
We quantify the controllable trade-off's in Fig. Panel (a) increases A, = X, at X 0, which reduces the PDE residual while also reducing diversity in the inferred permeabilities (measured via the complement of the mean pairwise SSIM; see App. E.II. Panel (b) fixes X₂ = X 20K and varies Xy. reporting MMD,, between the fine-tuned samples and the base dataset. As expected, stronger regularization preserves distributional fidelity (lower MMD) but yields higher residuals.
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These ablations illustrate how practitioners target residual reduction distributional fidelity by tuning (Ag, Xa Ay).
Rweak (rel)
SSIM Diversity a
Rweak (rel)
MMD
0.987
0.055
3
0.951
0.042
2
0.914
0.029
I
0.878
0.016
0
0.842
0.003
1k
10k 100k
1M
10M
base
0.1
(a) A and Aa vs. residual/diversity
(b) Regularization Ay vs. residual/MMD
Figure 3: Darcy ablations. (a) Increasing A = Ao with ' 0 lowers the PDE residual but reduces diversity in the inferred parameters (reported via SSIM-based diversity). (b) Sweeping X trades PDE residuals against fidelity the base distribution (MMD2). Each point averages 256 samples with shared noise seeds across settings.
Computationally, adaptation is lightweight: fine-tuning on noisy Darcy requires only 20 gradient steps (hyperparameters in App. E.3) and completes in under 15 minutes on single NVIDIA L40S, after which sampling proceeds at base-model cost with inference-time adjustments.
4.2 GUIDANCE ON SPARSE OBSERVATIONS
In many realistic settings dense observations of a state variable are available for pre-training a gener- ative model, whereas only few measurements of the latent parameter be collected. To sample from the posterior of parameter-state pairs that respect such sparse evidence we steer the genera- tive process through guidance. [Huang al. (2024) demonstrate guided sampling towards sparse observations from a model that was pre-trained on the joint parameter-statt distribution. Our ap- proach applies similar guidance mechanism, however, to model that was pre-trained on noisy state observations alone. We state details on the guiding mechanism in E.4 Figure g shows that the guided sampler adheres sparse measurements while preserving realistic variability in the gen- erated samples. Additional results for different amounts of conditioning observations are given in Appendix F33.5
Three samples through guidance towards sparse observations (white markers in right panel) of the permeability, showing plausible conditional distribution.
Figure 4:
4.3 LINEAR ELASTICITY
We consider alane-strain linear elasticity on Ω [0. 1]2 with spatially varying Young's modulus a(E) and fixed Poisson ratio. Boundaries are Dirichlet: left/right clamped, top/bottom receive in- ward sinusoidal normal displacements with zero tangential slip. During fine-tuning, we impose a modified lower-boundary amplitude to induce controlled misspecification (see App. B and include an MSE boundary penalty in the weak residual. We report quantitative BC results as the MSE at the boundary in Table 4.3 and qualitative comparison is provided in App. F] Our method attains low weak/strong residuals while keeping distributional shift modest; PBFM and FM+ECI drift distribu- tionally present high residuals (full details and non-curated samples in App.
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Model
BC error (MSE)
Reask (rel) 4
Ratrong (rel)
MMD,
MMD
6.98 x 10 (#: 0.53)
1.59 10 (1 0.37)
1.83 10 (#: 0.66)
0.24
PBEM
2.32 x 10 (+: 0.87)
6.32 x 10 (+ 0.82)
4.22 x 10
(+ 0.26)
0.92
0.54
FM+FCI
0.0
1.01 10
(# 0.13)
2.49
102 (+ 0.32)
1.16
0.36
Ours
1.71 x 10 (#: 0.50)
6.15 x 10" (# 0.77)
3.79 x 10 (# 0.87)
0.15
0.12
Table I: Linear clasticity under BC misspecification. Quantitative boundary-condition (BC) error, relative weak/strong residuals, and distributional metrics. Our method achieves low residuals with limited distributional shift.
4.4 HELMHOLTZ
We consider time-harmonic wave propagation governed by the heterogeneous Helmholtz equation Au (1 i tan 6) xux))2u on f [0, 1] with Robin boundary conditions. Training data use a small damping term (tan å > 0) producing complex attenuated fields, while fine-tuning assumes the idealized lossless model (tan 8 0), inducing controlled model mismatch.
Model
Criterion
Rweak (rel) 4
Raning (rel) 4
MMD,
MMD
FM
1.5 10 (+: 0.59)
2.55 10! (= 0.55)
0.18
0.03
PBFM
8.33 x 10 (4: 3.04)
1.22 x 10
(= 0.33)
0.09
0.03
Base AM
Rweak MMD,
4.9 10" (# 1.85)
1.34 10'
(= 0.32)
0.15
0.04
Base AM
5.64 x 10 2.09)
1.59 x 10!
(= 0.33)
0.13
0.04
Basc AM V
Rwosk
4.99 10 (#12)
1.16 10'
(= 0.33)
0.13
0.05
Base AM + P
MMD,
5.46 10" (#: 1,94)
1.59 10'
(= 0.33)
0.12
0.04
AM
Rwesk
4.3 x 10" (d: 1.29)
1.14 X 10 (= 0.29)
0.07
0.04
AM
MMDs
4.32 10 (# 1.43)
1.05 10 (= 0.30)
0.06
0.04
Table 2: Helmholtz: residuals and distribution (representative configs). Normalized weak/strong residuals and MMD metries. We include AM variants and PBFM-style baseline.
Table 2 reports representative configurations for each method, selected as either the setting with the lowest weak residual (Rweak) the lowest MMD,. Full results are provided in App. F The base FM model shows the largest weak and strong residuals due to the damped-vs.
mismatch. PBFM substantially reduces both residuals relative to FM and, notably, also lowers MMD,, and preserves MMD,.. o. The AM ablations (Base AM and Base AM+p) further reduce the weak residuals into the range 4.9 109-5.6 10". with strong residuals similar to those of PBFM, but they incur moderate increase in MMD, and MMD, compared to PBFM. Our full joint AM model achieves the lowest residuals overall (weak residuals down to 4.3x 10 and strong residuals near 1.05x 10') while simultaneously attaining the lowest MMD,, among all methods and maintaining MMD, comparable to the ablations. This indicates that the joint flow most effectively resolves the misspecification while preserving distributional fidelity.
4.5 STOKES LID-DRIVEN CAVITY
We consider steady incompressible flow in the Stokes regime (linear, low-Reynolds-number proxy) governed by - (v(x) + S, V u 0 with no-slip walls and a smooth moving lid. The dataset uses nonzero Kolmogorov forcing / o O, while fine-tuning assumes ʃ = 0, creating a systematic model mismatch. Figure reports the lid-driven cavity. We show only the Base AM variants and our joint model: the base FM model exhibits extremely large residuals (3.05 102 3.16) and omitted for clarity, while PBFM fails to converge to meaningful velocity-pressure fields (strong residuals 1.15 10 + 0.05; see App. In contrast to Darcy and Helmholtz, the attainable weak residuals of all remaining variants are similar (Rweak 4-15.. However, the joint model reaches substantially lower parammter-diistriuutio discrepancies, achieving MMD, x 0.07-0.13, whereas both ablations remain around 2.22-.2.. Overall, although residual levels are similar across AM variants, only the joint model can enter the low-MMD regime-particulane particularly for MMD,,. This highlights the joint flow's s greater flexibility in achieving high-fidelity parameter distributions.
trade-offs for the Stokes

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40
Base AM
Base AM
Rweak (rel)
30
R Reak (rel)
30
AM
20
X
20
M
14
10
I
0.2
0.4
0.6
0.8
1.0
0.10
0.15
0.20
0.25
0.30
0.35
MMDx
MMD
trade-offs. Weak residuals Rweak versus (a) MMD, and (b) MMD,, Across all variants, attainable residuals are comparable, but the joint model (green) reaches much lower parameter-distributie discrepancies (MMD,, A8 0.07-0. 13) than the Base AM (bluc) and Base AM+p (orange) ablations, which remain around 0.22-0.28.
Figure 5: Stokes lid-driven cavity:
4.6 NATURAL IMAGES: PARAMETRIC COLOR TRANSFORMATION
To demonstrate cross-domain utility, we apply our method to natural images by introducing a para- metric recoloring pathway: analogous the hidden PDE parameter, a here specifies a polynomial color transform that operates outside the latent space, enabling exploration of image appearances not well supported by the base distribution. We use a class-conditional Latent Flow Matching (LFM) model (Dao et al.) 2023) pre-trained on ImageNet-Ik k (Deng et al. 2009) and optimize PickScore (Kirstain et al. (023) with globally fixed prompt. As concrete example, we fine-tune on the class macaw with the prompt "close-up Pop Art of macaw parrot," yielding the samples in Fig. o Joint fine-tuning with recoloring produces markedly more vibrant palettes and, crucially, joint adjustments (e.g., background textures that the recoloring exploits). Details about the recoloring parametrization are given in Appendix E.7 and further non-curated samples are provided Appendix F.3.6
(a) Vanilla Adjoint Matching
(b) Joint model with parametric recoloring
Figure 6: Fine-tuning of LFM model on macaw class using prompt "close-up Pop Art of a macaw parrot", comparing vanilla Adjoint Matching with our joint approach.
CONCLUSION
We have introduced a framework for post-training fine-tuning of flow-matching generative models to enforce physical constraints and jointly infer latent physical parameters informing the constraints. Through a novel architecture, combined with the combination weak-form PDE residuals with an adjoint-matchig scheme our method can produce samples that adhere to complex constraints with- out significantly affecting the sample diversity. Experiments across PDE problems demonstrate the potential of this method to reduce residuals and enable joint volution-paramedd generation, support- ing its promise for physics-aware generative modeling. Future steps include adaptive approaches to optimizing trade-off between constraint enforcement and generative diversity, and extending the framework to more complex and multi poytiis systems, including coupled PDEs and stochastic or chaotic dynamics. We would also explore how this methodology be leveraged for uncertainty quantification and propagation, and downstream tasks such as optimal sensor placement and scien- tific discovery workflows.
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REPRODUCIBILITY STATEMENT
We report datasets, model backbones, training schedules, loss definitions, evaluation metrics, and the key hyperparameters required reproduce our results in the main text and appendix. Remaining implementation choices are documented in the released configuration files, We fixed random seeds where applicable and specify hardware/software versions.
LCKNONTESSEMENTS
This work was supported by funding from the German Federal Ministry for Education and Research (Bundesministrrium für Bildung und Forschung, BMBF) under grants 16IS24071A IA / 161S2407 and 01IW23005. SF additionally acknowledges support by the DFG through FOR 5359 (II) 459419731), TRR 375 (ID 51 511266698, and SPP 2331 (441958259, 553345933, 466468799). and by the Carl-Zeiss Foundation through the initiatives Al-Care and Process Engineering 4.0. We thank the anonymous reviewers for their comments.
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A USE LARGE LANGUAGE MODELS
We employed large language models to polish the manuscript (wording, grammar, and synonyms) and to assist with implementation details, including plotting scripts and code rewritingrefactoringg The research questions, problem formulation, algorithmic design and experimental methodology, however, were conceived by the authors. All LLM-produced text and code were reviewed, adapted, and verified by the authors prior to inclusion.
B DATASET DETAILS
In this section we detail the datasets used throughout our study. Our guiding principle was to select scenarios in which the underlying parameter fields contain sharp discontinuities, thereby inducing rich, non-linear behavior in the associated state variables and making the inverse problem decidedly nontrivial. Although the Darcy-flow benchmark follows the setup of Li et al. (2020). we regenerate data so that the sample count, grid resolution, and ground-truth parameters can controlled precisely. Complete scripts for producing all data will be released to facilitate transparency and reproducibility.
B. DARCY FLOW DATASET
The dataset comprises 20,000 independent samples, each pair (a, u) on the unit square Ω Ry the permeability and f -> the steady-state pressure solving
1]2
where a
V (a(E) Vu(g)) = =({), g = (61, E E
with homogeneous Dirichlet boundary conditions w/an and constant foreing f =
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Discretization. We use uniform 64 x 64 nodal grid with spacing Az 1/(64 1) and the stan- dard five-point finit--iiffernne scheme. Interface permeabilities are formed by two-point arithmetic averaging. Dirichlet values are imposed strongly, yielding sparse SPD linear system that is solved with a direct sparse solver.
Permeability sampling. We draw zero-mean Gaussian random field draw via cosine-basis Karhunen-Loéve synthesis on associated with the Matérn-type covariance operator
= (7 -
2, 3,
i.e,, using the DCT-II basis (Neumann eigenfunctions of a A) setting the DC mode to zero to enforce exact zero mean, scaling by the spectrum of C, and applying orthonormal inverse DCT to obtain a grid realization.
To model sharp contrasts, we threshold the Gaussian field into a piecewise-constant permeability,
Craw (E) < 0. {
12, Craw (E) 2 0,
a(g)
Observational noise. tive Gaussian noise
In experiments with noisy observations, we corrupt the pressure with addi-
= u00, e e~~10, I), a a= 10-
B.2 LINEAR ELASTICITY DATASET
The dataset contains N 10,000 independent samples on the unit square Ω [0, 1J2 Each sample is a pair (E, u) where E 52 - IR=u denotes the spatially varying Young's modulus and Ω - R² the plane-strad strain displacement field under fixed Poisson ratio
0.30. The boundary loading deterministic and identical across samples: the left/right edges are clamped, and sinusoidal normal displacements are prescribed on the top and bottom edges with zero tangential slip (amplitudes Atop Abot 0.10). For each E, solves the static linear elasticity equations (no body force)
V (E)) = 0,
=(EEv)) tr(e) (),
E
1/2 (V + Vul), EE
with Lamé parameters X(E,v) (1++)(1 20) and p(E,v) conditions
E 2(1+u)) and Dirichlet boundary
=(5i=0,55) {2)
=1 ==)====
uz =0) 0, Uy (51 E2 00) Abot sin {1)
Uy (E1, (2=1) = 0, uy (E1, =- = +. sin E1).
During fine-tuning, we set Abot 0.075 to enforce adaptation of the solutions, simulating a mis- specification between observed data and the assumed model.
Discretization. We use uniform 64 x 64 grid on Ω and standard recond-orded finite differences in the small-strain regime. Dirichlet data are imposed strongly. The discrete equilibrium equations are advanced by stable explicit umme-marching (damped gradient) scheme for the static problem until the global residual norm falls below 10 or a cap of 2 x 10 iterations is reached.
Coefficient field sampling (piecewise-constant Voronoi medium). Heterogeneous modulus fields E are obtained from Voronoi tessellation constructed by drawing a fixed number of sites uniformly in Ω, assigning each Voronoi cell modulus sampled log-uniformly within [1.0, 10.0), and rasterizing the resulting partition to the computational grid by nearest-site site labeling. This pro- duces niecewise-conster E with sharp jumps that emulate multi-phase media. To control interface smoothness. a separable Gaussian blur with standard deviation o 1.0 (in grid units) is applied to the rasterized field.
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B.3 HELMHOLTZ WAVE PROPAGATION DATASET
The dataset consists of N = 10,000 independent samples on the unit square Ω = [0, 1] Each sample is pair (c,u), u), where e: Ω IR denotes the spatially varying sound speed and u S2 C is the iime-harmonic acoustic pressure. For fixed angular frequency w > 0, the field i solves the damped Helmholtz equation
-Au(E) (1 i tan 6) A(E)² w(E) s(E),
x(E)
w
& S
c(E)
subject to homogeneous Robin boundary conditions
0,, + i eu{) = 0, { E 80,
with fixed admittance parameter > 0 and loss tangent tan 5 2 0.
Discretization. We use uniform 64 64 nodal grid on Q with spacing Ar 1/(64 1). The Laplacian - is discretized by the standard five-point finiteddifference stencil in the interior and homogeneous Neumann conditions the normal direction on all sides, implemented via ghost-point elimination. The Robin condition 0, + iyy = 0 is enforced by adding purely imaginary diagonal contributions at boundary nodes to the stiffness matrix. consistent with the underlying Neumann- type discretization. The heterogeneous reaction term -(1 tan d)x²u is assembled pointwise the grid. This yields a sparse complex-valued linear system, which is solved for each sample using direct sparse solver.
Sound-speed sampling. The sound speed c derived from a Gaussian random field constructed in the same manner as for the Darcy dataset, but interpreted as wave speed. We first draw zero- mean Gaussian field on the grid using a cosine-basis Karhunen-Loove synthesis associated with the Matérn-type covariance operator
= A)-
a T == 6,
using the DCT-II basis on St zeroing the DC mode, scaling by the spectrum of C. and applying the inverse DCT to obtain g. To model sharp material contrasts, map g to two-level sound speed,
Cmax: g(5) 2 0,
c(E)
=
Cmin: g({) < 0,
with Gmin = 0.8 and Cmax 1.2 in the reported experiments.
Source and damping. The source term is fixed, smooth Gaussian bump centered in Q,
(()) = exp (0.5. 2
0.05,
shared across all samples. We use a fixed angular frequency w 20 and Robin parameter y 0.03. Data is generated with small but nonzero loss tangent tan d 0.02, leading complex-valued damped solutions.
B.4 STOKES LID-DRIVER CAVITY DATASET
The dataset consists of N 10,000 independent samples on the unit square Ω |0, 1J2. Each sample a pair (N, u). where v : Ω - Ry denotes the spatially varying kinematic viscosity and Ω IR² is the steady incompressible velocity field with associated pressure p Ω R. For each v. (u. p) solve the variable-coefficient Stokes system
- -V Vu(E)) + Vp(E) f(E),
V V. u({) = O. EEN,
with smooth lid-driven cavity boundary condition on the top wall
x(11,2=11 = (Una sin² (nt Ei), 0).
and no-slip conditions on the remaining walls
<(E1,<=-0) 0,
u(fi=0, <2) 00 u(Ei=1, (2)
To excite richer flow structures, we add Kolmogorov-tyy body force E2) = Fo (sin((T E2), 0).
with fixed amplitude Fo 0.
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Discretization. The Stokes system is discretized using FEniCS (Baratta et al. 2023 Scroggs al. 2022b at AAlaas et al., (2014) on a conforming triangular mesh obtained from a (64-1) (64 gnd of squares, each split into two triangles. We employ Taylor-Hor Hood elements (vector-valuet Pa for u, scalar P1 for p) on the mixed space V x Q. The weak form reads
Ja "Vu: Vu: Vuda - -fo V.. .V-udr
Jo f.
VV
for all test functions (v,4) 4) E V x Q. with the lid and no-slip boundary conditions imposed strongly. For each sample, the resulting saddle-point system is solved with direct solver, and the solution (u, p) is sampled on uniform 64 x 64 grid on S2; the pressure is post-processed to have zero spatial mean in order fix the gauge.
Viscosity field sampling. The viscosity is obtained from Gaussian random field constructed in the same manner as for the Darcy and Helmholtz datasets. We first draw zero-mean Gaussian field y on the grid using a cosine-basis Karhunen- synthesis associated with the Matérn-type covariance operator
C (
a= 4, T= 6,
using the DCT-II basis on St zeroing the DC mode, scaling by the spectrum of C, and applying the inverse DCT to obtain g. To model sharp viscosity contrasts, we map g to two-level field,
= Ummxx: g(() 2 0,
((5)
Vmnin
g(E) 0,
with Vinn 0.02 and Vmax 0.5 in the reported experiments.
Lid and body-force parameters. In all samples we fix the lid velocity scale to Una = 1.0 and the Kolmogorov forcing amplitude to Fo = 2.0.
C PRE-TRAINING OF FLOW MATCHING MODELS
We adopt the vanilla Flow-Matching (FM) procedure Lipman al. (023)) Based on the refer- ence flow
X, 3,X, Xo
with B, = and 7i parametric model u0( t). Given an end-point D1 ~ Pdata the conditional vector field is available
can define the conditional vector field as training targets for
1
(x[<<])) = --1 1- (I1 xx)
x).
This leads to the simplest form of Flow Matching objectives:
CPM El t) (Xi - -xoil
where Xo W(O, X1 Pdam and 1).
Network Backbone. For PDE data, the mapping is realised with a U-FNO (Wen et al. 2022 which combines Fourier Neural Operator (FNO, [L et al. (020)) with U-Net (Ronneberger al,, 2013) layers. The FNO layers have an inductive bias towards low-frequency solutions and are therefore particularly suited for modeling PDE data, while the U-Net layers help to refine outputs and produce discontinuities.
Departing from the original design, we use the U-Net skip-connection structure employ more powerful cine-conditioning in the U-Net layers by using FiLM-style Adaptive Group Normalization (AdaGN) time conditioning, i.e., predicting per-channel scale and shift from sinusoidal time embedding and applying them after GroupNorm in each residual block. This follows the seature-wise linear modulation idea of Perez et al. (2018) and its diffusion U-Net instantiations in Dhariwal & Nichol 202T); Rombach et al. (2022).
layers. Also,
We prepend the physical input with fixed sinusoidal encodings for absolute spatial coordinates and the normalized time stamp the spectral backbone receives explicit space-time context. Padding
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inside the U-FNO (for the spectral convolution layers) is reflective, which has empirically worked better than replicative. Before training, data standardised to zero mean and unit variance.
Table 3 compiles the hyperparameters of the U-FNO architecture used for the base FM models in our experiments.
Table 3: U-FNO (2D) backbone hyperparameters used in our experiments.
Hyperparameter
Value
Description
Number layers
4
Count of spectral and U-Net operator blocks.
Padding mode
re rlect
Boundary padding for convolutions/lift.
Input channels
1 - 3
Channels of input field x.
Output channels
1 - 3
Channels of output field.
Time embedding dim
32
Dimensionality of time conditioning. Dimensionality of coordinate embedding.
Spatial embedding dim
32
Lifting width
256
Channels in input lifting/projection. Retained spectral modes per axis.
Fourier modes (kx, ky)
[32,
321
Spectral block width
32
Channel width within FNO layers.
U-Net base widths Embedding width (U-Net)
32
Stage-wise channel widths.
64
Channels for conditioning embeddings. Stages with self self-attention (empty none). Multi-head count if attention enabled.
Attention stages
[]
Attention heads
[]
Total number of parameters
19
On natural image data, we use pre-trained Latent Flow Matching Model (Dao et al. 2023) based on DiT (Peebles& & Xie 2023) backbone.
Optimization. We train the FM backbone with AdamW (Loshchilov Hutter 2017 using linear warmup of the learning rate from 0 the base value over the first Pwu fraction of training steps, followed by constant learning rate thereafter. For evaluation stability, we maintain an exponential moving average (EMA) of the network parameters (EMA 33 0EMA + (1 - B) 0, practice rooted Polyak averaging (Polyak & Juditsky 1992) and widely adopted in modern deep models (e.g., Tarvainen & Valpola 2017).
Table Flow Matching (FM) training hyperparameters and schedule.
Hyperparameter
Description
Batch size
128
Minibatch size per step.
Base learning rate
1e-4
AdamW step size.
Warmup fraction Pon
0.01
Fraction of total steps used for linear LR warmup.
Epochs
300
Full passes over the dataset.
Optimizer
AdamW
Applied FM backbone parameters.
Weight decay
0.01
AdamW 1.2/decoupled decay coefficient.
EMA decay
0.9998
Exponential moving average of weights for evaluation.
Schedule after warmup
constant
LR held constant after warmup.
The FM training configuration is summarized in Table For the Darey dataset (20k samples), training takes around 12 hours single NVIDIA RTXA6000 GPU. For the smaller datasets of 10k samples but with more channels (Elasticity, Helmholtz, Stokes), training takes around hours using the same configuration.
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D METHOD: IMPLEMENTATION DETAILS
In this section, we provide further details on the implementation of our fine-tuning method. This includes neural network architectures, the specific design of test functions for the computation of the weak residuals, numerical heuristics, and finally the full training algorithm for fine-tuning. All relevant code is implemented using Python 3. 3.12.3, specifically, neural architectures are imple- mented in PyTorcc Paszke et al. (2019) version 2.7.
D.1 INVERSE PREDICTOR
We parametrise the inverse map yp with two-layer U-FNO, mirroring the spectral-spatial bias the forward backbone. However, we increase the capacity of the U-Net components and use attention the two lowest resolutions (stage indices 2 and 3). Since the inverse predictor only operates at t 1, we drop the temporal but keep the spatial conditioning, Exact parameters are stated in Table
Table 5: U-FNO Inverse Predictor.
Hyperparameter
Value
Number of layers
2
Padding mode
re flect
Input channels
1 - 3
Time embedding dimension
0
Spatial embedding dimension
32
Output channels
1
Lifting width
256
Fourier modes (kiz, ky)
[32, 32]
Spectral block width
32
U-Net base widths
[64, 64, 96, 128]
U-Net embedding width
64
Attention stages
[2,
Attention heads
4
D.2 ARCHITECTURE MODIFICATIONS
Fine-tuning augments the base U-FNO map x + buse (x, t) to a joint, a-conditioned vector field
(v "(,a) (x,
implemented as residual corrections around U-FNO core. Given the padded input Tpad (with time/space embeddings) and conditioning fields and base the base feature stack x. produced by the original skectral+skipp+U. U-Net blocks. A first correction head (a U-Net) takes [x., 0 a] and outputs additive refinement, yielding
base
Ut.z + Us(2,, base a, t).
After unpadding, a pixel-wise correction (lighteeight channel-wise MLP) further adjusts of using local features and 2D positional channels,
oft- +- of_ Mo(pos(I...) t), of..).
which provides extra capacity for rapid adaptation without altering the global operator. For the parameter dynamics, we adopt a strictly residual strategy that conditions on both a and the baseline field:
n = hase + Uc(Ipad O, Ut
base
vt
where U4 is second U-Net. All correction heads zero-initialized at their final projection layers, so that at the start of fine-tuning n (uhase v and departures are learned smoothly. Table lists the parameters of the correction U-Nets.
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Table 6: Parameterization of the fine-tuning U-Net heads Ug and Ua.
Hyperparameter
Value
U-Net base widths
[64, 64, 96, 128]
Embedding width (time/aux)
64
Hidden lift (internal width)
256
Self-attention stages
12, 3]
Attention heads
Overall, the modifications to the base architecture add around 6M parameters to the model, resulting in total 225M parameters.
D.3 WEAK RESIDUALS AND TEST FUNCTIONS
Darcy Flow. A pressure field that solves our Darcy flow equations fulfills
-V- (aVV - f= L.u 0 on Q c IRd with homogeneous Dirichlet data u|an = For
any vo E Co (2) we compute the L inner product by multiplying with & and integrating:
(Lau, (-V-
(-V-
vb de
One application of the divergence theorem yields
/
(Cou, () = Jon
Vu. vv) 1/ de
(a Vu- Vτ)) - f de,
where the boundary term vanishes because of tl = 0. This expression only contains a first-order derivative of and be used to compute in the computation of the weak residual. To obtain dimensionless, coefficient-scaled quantity comparable across locations, we normalize by the local mean permeability over the support of vb.
/ Supps a(E) de,
R(ya] Z
R
supp 2
In practice, supp v is the compact patch where ab (or its mollified variant) is nonzero, so that c. captures the local coefficient scale used normalize the residual.
Linear Elasticity.
For any compactly supported vector test vi E (9;; the weak residual is
R (Cgu, ()))(00)
-( dg.
single integration by parts yields
R
/
(o [
V.d oo:
[ lo:Vidt de,
V
since vlag = Here "a Val" is the Frobenius product and we denote the stress components by Ozz: Ozz:Ozy Twz), yy. To reuse the same scalar test generator for both momentum equations,
restrict to tests that share a single scalar profile in both components. Coneretely, we take
(x) (vb, 0):
g5(8)
(0, vb),
/ ().)..
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With this restriction, residuals can be computed component wise
(Cgu, (Cgu,blay) - [ (oen og, v + aay 05)45 ds.
(Ceu, (blu) = (0zg 02, ow 0c,05 de.
To obtain a dimensionless, coefficient quantity, we normalize by the local mean modulus over the support of vb,
(Ceu, 2b(m)) F.p.
R(0) [bb] (CEu, eblo)) F.
E supp v supp V
E(S) de,
[+]]
For family of tests {uk]k= the scalar residual used in experiments the lp aggregation over components followed by averaging over tests. Therefore, the weak residual for the clasticity experi-
ment is
N Σ (R()) (
Rewaak =
N
Helmholtz. For the Helmholtz exprriieetssssssssss we consider complex-valued time-harmonie pressures w: Ω C satisfying
L.u := (An(E) (1 i itan 8) m(E))u(() - s(E) = 0, <<{) = c(E)
w
where c(E) is the sound speed, > 0 the angular frequency, tan 5 2 0 the loss tangent, and s({) fixed source. Writing u un + iu, with real and imaginary parts ug, ur Ω R, we split the operator into its real and imaginary components
C (c. ( cc( us -) + +i((! c(u
with
(R)) un = Aun - K 2
c == - Aus x2 x²u1,
C( us == as
clu)g= == aajuR
where an == x² and a, := - tan 8 n For any scalar test function và Co. (S2), we define real and imaginary weak residuals as
R. ( == (c( un + c( us s, w) Lo(D)'
R0) := (c c(U v) Lo(0))
single integration by parts in the Laplacian terms gives
Jmn (VuR n) ds d + / (VUR- - aRuRt arupp (b) de. /
R ( [b]
R [1] = - (Vui (Vu- n) v ds + (Vus Ve - a dg.
Since v compactly supported in the interior, the boundary contributions vanish and we use only the volume terms in the residual. As in the elasticity case, we reuse the same scalar profile vi for both channels (real and imaginary parts) and evaluate R. (())(()) and [1) on identical compact patches to increase computational efficiency.
To obtain dimensionless quantity that is comparable across locations and coefficient scales, we normalize by a local Helmholtz energy associated with the bilinear form
/ Suupps s
(u)
Vuull ( Vuul + x{E)² (ux(E)² + un({))) de,
which combines gradient and reaction contributions both channels. The normalized residuals are (() = R0
R( [+] = (R)
(u)
Given a family of scalar tests ( the scalar weak residual used in experiments is obtained by aggregating real and imaginary components and averaging over tests,
Rweak = (xx)) ((xe)))
N
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Stokes lid-driven cavity. For the Stokes experiments, each sample contains a Selocity-pressieee pair (u, p) solving the variable- viscosity incompressible Stokes system
(u( p) = (- (vVu) + Vp, V. u) = (0.0) in Ω
with v:: > Ryo denoting the viscosity field. Writing u = (Wz, uy). the operator decomposes into
(u, V. (vVug) + Ps,P, (u, V (vVug) + de,P. (idiv) (u,p) = 0g ux + 052 "y
For any scalar test function & E (9), we form momentum and incompressibility residuals by testing each of these three equations with the same scalar profile,
R [+)
(c. (u.p), v) L2(S))
R = (u, p), v) 12(0)'
[ = (c.dir) (u, (u,p), 1b) 1.(()))
As in the Darcy and elasticity settings, one integration by parts isolates only first derivatives of the solution inside the volume integral. Since v/ has compact support inside Ω all boundary terms vanish and we obtain the local weak forms
R ( / ((u (V V (0g,p) 1b) de,
R (1)) [vb]] = S/1 (uy, Vú + (de.P) v) de,
R (div)) [+b] = / (0s, Usr + )y 1b de.
To obtain dimensionless, coefficient-scaled residual comparable across locations, normalize by the local viscosity-weighted kinetic energy over the support of v),
(u) == [ supp ot (2(E) (Vv((E))) + [l)(E)))) de.
where Vu 1/-
10g 2z + 10 Ux 12 + 10g uyl + 10 upl 2 The normalized weak residuals are R(0)(v)
R. w
R(div) [+b]
R
R [+b]
(dd
/ (u)
/ (u)
(u)
Finally, for collection of scalar tests {vx \N k= the scalar weak residual used in experiments aggre- gates momentum and incompressibility contributions and averages across tests:
N
Σ (R()) + (vx)) + (R (dit)) (bal)) N
1
R Rweak
Wendland-wavelet test family. For estimating the weak residual accurately and to provide a strong learning signal, we need to sample sufficiently many test functions. Evaluating them the entire computational grid would be prohibitively costly. We therefore consider test functions which are locally supported, such that can restrict computations to smaller patches. Wendland polynomials are a natural candidate meeting these requirements since they are compactly supported within unit radius and allow for efficient gradient computation. Here, will describe the test functions in detail.
Each test function is drawn from a radially anisotropic family.
L (x) (1 r(=)) (4r(z)++ (1- (1- (l-64br(a))
Wendland C2
optional wavelet
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where r(z) = ( Length-seales oj are uniformly sampled from the range [mmin j. mmx A3}- By multiplying Omin and mmax with the grid spacing A, of axis j. we ob- tain parametrization that is intuitive to tune since the length-scales of the test functions given in pixel units. Instead also sampling center points c uniformly and independently from the full domain Q2 instead start from the grid coordinates of the data, considering each grid point center. We found that this way of ensuring spatial coverage improves training efficiency. To still retain stochasticity, apply i.i.d jitter each center point. Ber(p) randomly toggles a high- frequency wavelet factor that provides additional variability within the test functions and proved especially effective noisy data.
To enforce vasaan = O, we multiply every test function by a bridge mollifier
m(g) ((E Emin (Emax - /)) (fmax Emin)?
applicd per axis. This legitimizes the application of integration-by-pat in the derivation of the weak forms. At training time we sample set of test functions {2b(() New independently per residual evaluation and define the loss
)
Rweak (z, a)
E(Crr 2211)
Nesst
D.4 ADJOINT MATCHING DETAILS
As mentioned the main paper, we introduce a scaling coefficient i: that allows us to attenuate the magnitude of the noise variance. In this section, provide justification for using > 0 and lay out numerical heuristics used in the implementation.
Memoryless Noise. Domingo- Enrich et al. (0255) define generative process with noise schedule o² (t) to be memoryless, if and only (t) 27) x(t) with X [0. R chosen such that
(- X(s) 2
Vt E (0.1) lim By exp
ds = 0.
Specifically, they refer (() 2nl the memoryless noise schedule. In setting of Be I. and 7/- = 1 the memoryless noise schedule can be simplified to
(t)) 2(1
Lemma (Scaling of memoryless noise). Consider a generative process as in 7 with Br = L and % For 0 < the schedule 0²(t) (1 (1- K) 2mp memoryless.
Proof. First, consider the integral term. The desired o² o²()) - x) 2n implies that
x(1)= -kmm -27
1(1 - 1)
With this, simplify the integral term:
[ x(s) ds ( 1 ds K Jr 1 1- ds
2
s(1 ((1s))
[log(s) - -log(1 8)]
t log t'
log 1-t
t!
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Thus, for arbitrary fixed E (0, 1), it holds that
By exp /"" X(s) 27
t'
ds
l' exp
(log
log
1-1
11
t
t'
-x
t'
K
0.
v'+0
const
-0
brundled
Therefore, scaling down the noise schedule by a factor 1 - K. is justified and consistent with the theory provided in Domingo-Enrich et al. (2025).
Heuristics. Still, the term causes numerical problems for t = 0. Furthermore, it forces the control be close to zero for close Following the original paper, we instead use
h t+h
7
where choose h the step size of our numerical ODE/SDE solver. This resolves infinite values and allows for faster fine-tuning by letting the fine-tuned model deviate further from the base model close to
While x is an effective tool mitigate residual noise in final samples, increasing slows down train- ing. As another way of improving sample quality without adding computational cost, we consider non-uniform time grids when sampling memoryless rollouts. We observe that the most critical phases of sampling are at the beginning, where noise magnitudes are the highest, and at the end, where final denoising happens. Therefore, we tilt the uniform grid towards the endpoints:
Let S E N be the number of steps and define the uniform grid t; = i/8 for i O,...,S. S. We tilt this grid toward the endpoints by first applying a cosine ease mapping
g(() = - cos(TT)), E [0, 1],
which has s(0) 0, (1) For mixing parameter & [0, 1). the tilted times the convex combination
t) + (g(()). i=.....
{ is strictly increasing and distributes grid points more densely near 0 and for 0. For PDE experiments, we use = 0.9. This heuristic was not needed for latent-space models.
Loss Computation. As in the original paper, do not compute the Adjoint Matching loss (Equation 4) for all simulated time steps, since the gradient signal for successive time steps is similar. Note that for solving the lean adjoint ODE, we do not need compute gradients through the FM model. therefore we can compute the lean adjoint states efficiently but save computational resources when computing the Adjoint Matching loss. Again, the last steps in the sampling process are most important for empirical performance. For that reason, we also compute the loss for fraction of last steps Kiss. Additionally, we sample K steps from the remaining time steps.
To ensure stable learning, we apply clipping function to the loss to exclude noisy high-magnitude gradients from training. Empirically, the values provided in Domingo-EEnrch et al. (2025) work well our setting, i.e. we set the loss clipping threshold (L.CT) as LCT, 1.6 X and L.T. = 1.6 X respectively.
D.5 FULL TRAINING ALGORITHM
Algorithm I details the complete optimization loop used in all experiments. Starting from the pre- trained base flow basc attach two residual heads that (i) condition the state flow of the latent
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parameter and (ii) predict the parameter flow v²..- Their respective output layers initialized to zero. The inverse predictor oo is first pre-trained on base samples and then frozen, providing surrogate target flows for Each epoch rolls both the base and fine-tuned trajectories the tilted time grid, solves the lean-adjoint equation, and updates only the fine-tune parameters 0 through the udjoint-Matchinn loss.
Algorithm Adjoint Matching with Joint State-Paramete Evolution
Notation: base fftt E denote functions (neural vector fields). Evaluations time are written in base
wasee ( l). bold, e.g. Nx,t = oz
Input: base state field u initialize et ubase and add heads for eff oft
Pretrain o on xı samples from hase by minimizing wenek (2 ( (xx))
2: Freeze base and yo; let 0 be the trainable parameters of n".
3: for epoch .....
4: Sample zo (N, 1) and T) + GET (T).
5:
6:
for i
do
7:
ft t,, 1, h+ t+ t.
8:
Base state velocity:
9:
v base = w.ase (Ize 1). 2.
Surrogate terminal parameter:
11:
② = =x (1- v base
12:
a p())).
13:
Base parameter velocity (surrogate):
14:
base = (afe aa.)
15:
Fine-tuned velocities:
Vz. a (It; Ole, Vo,t
ft
17:
ftt
a.t = vaa (x',, Oiey a.t f).
base
18:
Forward steps (with noise (t))::
19:
24
base STEP(x buse base ht),
2t STEP(> vt o (t), ht),
21:
a = STEP(O vst ht),
22:
end for
23:
24:
(an,t, Gas)ee SOLVELEANADIOINT ((xbase) (xft {aft)).
25:
Subsample time indices: form I C (O...., L.. L} according to K and Lisst
27:
Per-step drifts: for each 4 E I compute
29:
b% ft + a(t,)² ft
[ E.f, 7,45 af:); b base vbase base + a(t;)² 2 2.6,
2n((,.] 2,1.
(vbase
base base
l,te 1tc
30:
a = ft o(t)) (v
a.t, ((taa) a,t; tst af:): b base
base
o(t))
busse F146 a
n(te)
31:
32:
Per-step losses (with clipping): for each t; E I set
33:
(t(+ = min (t,) -b... b base + o (t.) a,,,, LCT,
34:
(t( = min{ (t=) -bbft b base) + (t+) aa,t, LCT.
(squaring and clipping applied elementwise, then summed over spatial dimensions)
36:
37:
Adjoint-matching objective:
38:
L [ (L,e)(e((tt) (a()))).
39:
0 + GRADIENT.DESCENTUC
40: end for
41: return
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E DETAILS: EXPERIMENTS
Here we describe the fine-tuning configuration of the conducted experiments. Specifics about the base model FM training can be found Appendix [ In all experiments, use A as the optimizer with weight decay of 0.01 and all fine-tuning is conducted on single NVIDIA L40S GPU. As reported in the main paper, fine-tuning for Darcy takes less than one minute per epoch. For the other, computationally more expensive experiments, one epoch of fine-tuning takes around one minute (1-2 hours for 100 epochs).
E.1 METRICS
SSIM diversity. In panel (a) of Figure we report SSIM-based (Wang et al. 2004) metric for diversity, which is implemented as follows:
Given batch {a of single-channel parameter maps scaled [0, 1], define diversity as the mean complement of the pairwise structural similarity index (SSIM)
(ssim((ar)Ea))
(1 as) IsKjjsB
Σ
0s E [0,
B
With SSIM diversity.
[0, (data range 1), DSSIM
E [0, and larger values indicate greater sample
Maximum Mean Discrepancy (MMD). To quantify distributional differences between generated samples and the reference dataset pe we use the squared Maximum Mean Discrepancy (MMD²) with an RBF kernel (Gretton et al. 2012). Given two batches {x, C RD and (y C R., let
(u, v) = exp( 20 ||ue u1B))
20
The empirical MMD² is
MMD 2 (x, v)
Σ NO 2
(x,, +
Σ 3.
ko (us,
2 N1
Σ i
(I..))))
N2
In all experiments we use the median heuristic (Gretton et al. o² to the median of the pairwise squared distances across the pooled samples 2 = {or} {8g}. In practice subsample up to 1000 points to compute
2012), which sets the kernel bandwidth
0² = median I| 2b112 < b) +
= 10
which yields stable and scale-adaptive kernel width (Garreau et al. 2017)
E.2 ABLATIONS AND BASELINES
Single Adjoint Matching. To isolate the effect of modeling a joint flow over (x, (x), we include two ablations based the vanilla Adjoint Matching (AM) objective of Domingo-Enrich 2025). Both variants optimize the standard lean-adjoint control loss over the state trajectory [ using the same noise schedule, ODE discretization, and step subsampling as our full method.
al.
Base AM freezes the inverse predictor sp. It is used only to compute PDE residuals but receives no gradient updates, and no joint flow over is modeled.
Base AM+ keeps trainable and backpropagates through the residual term r. but still does not model coupled flow over (z,a). a). This variant tests whether updating the inverse predictor alone can stabilize residuals without introducing surrogate base flow.
As an alternative baseline we adapt Physics-Bii -Based Flow Matching (PBFM) (Baldan et al., 2025) to setting. Since PBFM requires paired (æ, a:) information during training, augment the method with our pre- pre-trained inverse predictor oo to estimate a and enable residual computation. Training follows the original ConFIG (Liu et al. 2024) update rule: each iteration we compute the
PBFM.
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FM loss and the residual loss, extract their gradients, and form conflict-free update via ConFIG. Our implementation mirrors the official code except for minor modifications to support complex- valued weights (arising from FNO backbones). Due to memory constraints, lower batch sizes than with vanilla Flow Matching (16 instead of 128). We chose the number of epochs such that the total number of update steps is twice that of our base FM models.
have to much
PBFM serves as a comparison against a training-time physics-regularized baseline, whereas our method enforces physics post-training (data-free). Note that all our PDE settings deliberately in- troduce mismatch between the physics assumed during fine-tuning and the physics underlying the training data. Such misspecification inherently challenging for training-time methods like PBFM which naturally places them at disadvantage.
E.3 DARCY
For the Darey experiments, we use the hyperparameters listed in Table Note that one epoch amounts to exactly one gradient descent step, meaning that we only fine-tune for 20 total steps. The parameters Az da and y are varied in the experiments, see Figure Bin in the main text.
Table 7: Darey fine-tuning hyperparameters.
Hyperparameter
Value
Time-tilting factor
0.9
Aa
varying
Xa
varying
Ay
varying
Kiast
20
K
20
Noise scaling K
0.9
Sampling steps (per trajectory)
100
Training epochs
20
Learning rate
0.00002
Batch size
15
E.4 GUIDANCE
For generating the guidance results, we use the same model as in the Darey experiments to further highlight that Instead of the usual Fuler stepping, here we use a Heun sampler following Huang et al.) (2024). While this is more expensive, since need to differentiate through two forward passes of model to obtain the guidance gradient, it empirically improved faithfulness to the sparse observations significantly. Note, however, that only guide on sparse observations and not PDE residuals. The full sampling procedure is shown in Algorithm In our experiments, we use 100 steps for sampling and guidance scales of 7/2 Ya 0.75.
infer functional joint distributions when starting from noisy observations.
E.5 ELASTICITY
Fine-tuning in the elasticity experiment is more challenging than the Darcy denoising experiment, which is why we increase the number of fine-tuning epochs to 100. Other parameters are the same as in the Darcy experiments.
We compare our fine-tuning approach with the inference-time correction method ECI presented in Cheng et al. (2024). Our implementation of the sampling correction method is given in Algorithm 3 For ECI, we set M = 5 and use 1000 steps when sampling, compared to 100 steps used when sampling from the fine-tuned model. The reported residual heat maps in the main paper show one test function per grid point, where the magnitude indicates local error without aggregating across test functions.
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Algorithm 2 Guided Heun sampler with sparse observations
Require: initial state zo; initial parameter og; observed targets aobsi index set steps S; guidance scales Tai base fields vi: of; fine-tuned joint field ei
Ensure: trajectories (x()) Ji=o and {a )i=0
1: [0) + To
2: (0) t- 0g
3: t; t i/S for i = .,...,S
4: h 1/5
5: for 1 t S do
6: Predictor (Euler):
7:
en t 1/ (x(²)
base t 1i (1(()))) a (1) 20))
8:
(J", (²) f- 1/ (x((1, a1
9:
10:
i + x + h
à al (1) + h
12:
if < S then
13:
Corrector (Heun):
14:
t ur buse, o/us (0, â, v
of
16:
(07, 04) &- 2 2 å, +
17:
One-step terminal extrapolation: â1 à + (1 t(+1) 50
18:
19:
Sparse-observation loss:
20:
L
Ejez aaae||aoo [oota - an 11
Heun average update:
h
22:
(1+1) - (+)
2
23:
a (++1) - (1) +
(Ta + 6
24:
Decaying guidance:
f / i/5
26:
((i+)) t x (1+1) 8/1 V 115) L
27:
(i+1) t (++1) s 8/a Vo
28:
else
29:
Last step (no correetiion/uiddanc): ((++)) (1+1) t à
30:
31:
32:
end if
33:
end for
return {2(1) Rea {a
E.6
HELMHOLTZ. AND STOKES LID-DRIVEN CAVITY
For both Helmholtz and the Stokes lid-driven cavity problems, we use the same configuration as in the elasticity experiment (see 8), training for 100 epochs. We only adjust the number of input/output channels.
E.7 NATURAL IMAGES: RECOLORING
For natural -maall-immeeeeee experiments, adopt the class-conditional Latent Flow Matching (LFM) model of Dao et al. (2023) trained on ImageNet- 1k (Deng et al.) (2099) with a DiT backbone (Peebles & Xie
2023).
We fix ImageNet class label condition the generator and hold global text
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Table 8: Elasticity fine-tuning hyperparameters.
Hyperparameter
Value
Tile-nilting factor
0.9
A
varying
Xa
varying
Xy
varying
Kinst
20
K
20
Noise scaling K
0.9
Sampling steps (per trajectory)
100
Training epochs
100
Learning rate
0.00002
Batch size
15
Algorithm ECI-style sampling with boundary correction
Require: initial state steps S; inner correction iterations M; model drift ve (+): correction oper- ator C(-); noise sampler Noise(.)
Ensure: trajectory (x(0) Ji=0
1: (0) 6- To
2: t, i/S for i O,....S
3:
for = to S do
4:
t 4
5:
grell + t++1
6:
Inner ECI corrections at fixed l:
7:
I + (())
8:
for j to M do
9:
- vx(
Pas + ++ (1 (1- t)v
one-step Euler update apply boundary correction
11:
Zeon
t C(Fcn)
12:
- Noise(shape of mq)
13:
+ (1 - 1) 17 ( Tewrr
stochastic convex blending
14:
end for
Final correction and roll-forward to gnext
16:
+ (
Ias ++ (1
18:
Toom + C(Ix))
-- Noise(shape of xo)
((++)) +- (1 (next) + neess From
21:
end for
22: return
(o (0) Ji=D
prompt to define the fine-tuning direction. As reward use PickScore vl, CLIP-H/14-basee preference scorer trained on the Pick-a-Pic dataset, that evaluates image- text compatibility via co- sine similarity the CL.PP embedding space (Kirstain et al. 2023) Radford et al.) 202T Cherti et al., 2023 Schuhmann et al. 2022). implemented with TRANSFOR MERS (Wolf et al. 2020).
Concretely, we maximize R ((T(((C ((x), a). c). where the latent-space decoder and T(,, a)) is parametric per-pixel recoloring with coefficients The recoloring operates in logit space avoid
29

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Published as a conference paper at ICLR 2026
saturation: with ((2) x = (-+
Ie(E)
((5) log (E
E IR
we apply a residual r(f; a (x))) built from RGB monomials up total degree and map back via y(E; = d(((E) + r((5:a) a)). Equivalently, for channel c {R, BR,
Ve(E:a) ((clt) +
Σ m=1
aem 6m (x(()) ac.u), Qm &D.
This residual parameterization is identity at initialization (a 0) and provides low-dimensional, channcl-coupled appearance pathway that can disentangle content from presentation or reach color- ings underrepresented by the base LFM. It is related to CNN-predicted polynomial color transforms (quadratic Chai al. (2020)), whereas we use cubic polynomials and learn solely via the reward.
The parameter predictor yp here maps from latent feature tensors to parameters First, com- pact scene descriptor is extracted by passing 2 through convolutional transformations, aggregating information across multiple spatial scales via downsampling and global pooling, and enriching with low-order channel statistics of z (e.g., moments). The concatenated descriptor is projected into fixed embedding space, refined by lightweight LayeroayrNorm MLP with residual connection, and finally mapped by linear head to the recoloring coefficients a.
Building on the inverse predictor, we augment the base DiT backbone with two lightweight residual heads that couple image dynamics and color-parameter evolution. First, the image path predicts the base drift t/ (x: t). Then, we form compact a-token by flattening the current color parameters a and mapping them through small MLP. This token is broadcast to hig -channe feature map and concatenated with (ar, base into shallow U-Net "correction" that outputs an additive refinement, yielding
base Utz
+ UL(2, base map(a), t).
V1=
=
Second, the parameter path updates the polynomial recoloring coefficients by residual on top of the baseline parameter field heese. Here, dedicated a-projection module mirrors the inverse predictor: multi-scale pooled conv features are fused-at the token level-with tokens from a and eta via small fusion MLP, AdaLN/FiLM modulation of a head token, and light SE rescaling of conv features. The head then predicts a delta added to the baseline,
buse U2
Uc(2z b/mee)
t,a
All correction heads are zero-initialized at their final projections, so fine-tuning starts from the base behavior and departs smoothly as training progresses. Table p lists the parameters the correction
U-Nets. In total, the modified architecture adds =9M parameters to the z= 133MM parameters of the base DiT backbone.
Table 9: Correction head hyperparameters (image and parameter paths).
Hyperparameter
Value
ka
16
U-Net base widths
[96, 128, 160, 192]
U-Net embedding
96
U Net hidden lift
256
Attention stages
[2, 3]
Attention heads
With the LFM model, we use 40 steps when sampling and use the same training specifications as the original Adjoint Matching paper (Domingo-Enrich al. 2025). Again, we first pretrain the predictor so, and then perform joint fine-tuning with our Adjoint Matching formulation. Fine-tuning performed for 100 epochs with a batch size of 15.
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Published as a conference paper at ICLR 2026
F ADDITIONAL RESULTS
In this section, we provide full experimental results and samples from the trained models.
F.1 FULL EXPERIMENTAL RESULTS
F.1.1 DARCY
Base AM
Base AM
Rweak (rel)
Rweek (rel)
3
3
AM
FM
PRFM
2
0.0
0.1
0.2
0.3
0.4
0.5
0.0
0.1
0.2
0.3
MMDx
MMD
80
80
Retrong (rel)
Rastonn (rel)
60
60
40
40
20
20
0.0
0.1
0.2
0.3
0.4
0.5
0.0
0.1
0.2
0.3
MMDx
MMD
Figure 7: Darcy dataset: scatter plots showing the relationship between PDE residuals (weak and strong) and distributional discrepancies (MMD,,, MMD,.) across all model variants and hyperpa- rameter configurations. Each point corresponds to one configuration; lower values indicate better physics consistency or distributional fidelity.
Table 10: Darcy dataset: full residual and distributional metrics for all methods and hyperparameter settings
Model
X
Rweak
Ratnong +
MMD, +
MMD, +
3.56
100 0.62)
x 10 3.20)
0.005
0.011
PBFM
1.69
10th 0.14)
1.64 x 10 1.66)
0.545
0.330
Base AM
1.89 10" (+: 0.40)
5.55 x 10 (+: 2.04)
0.238
0.057
Base AM
5k
1.84 x 10" (# 0.55)
4.22 x 10 (# 1.63)
0.362
0,094
Base AM
10k
1.95 10 (# 0.81)
2.95 10
1.14)
0.380
0.114
Base AM
20k
2.26 x 10 (#: 1.38)
2.31 10 (#: 1.10)
0.405
0.125
Base AM
100k
3.44 x 10" (d:2.
2.04 10 1.05)
0.507
0.145
Base AM
1M
3.68 x 10" (.17)17)
2.03 x 10 (1 1.04)
0.508
0.145
Base AM
100M
3.72 10" (.05)
2.02 x 10 (# 1.03)
0.512
0.146
Base AM
lk
2.29 10" (1 0.42)
5.32 x 10 1.89)
0.255
0.061
Base AM + yp
5k
1.68 x 10 (# 0.51)
4.01 x 10 (# 1,46)
0.376
0.095
Base AM
10k
1.85 x 10 0.66)
3.2 x 10' (# 1.16)
0.394
0.107
Base AM + GP
20k
2.15 10 (1: 1.18)
2.26 10 (L 1.00)
0.409
0.122
Base AM +
100k
3.36 x 10 (.61)
1.98 x 10a (1: 1.03)
0.481
0.143
Base AM + 6p
1M
3.82
10 3.27)
1.87 10t 0.99)
0.518
0.148
Base AM +
100M
3.58 x 10
(+: 3.06)
1.9 x 10 (+ 1.01)
0.519
0.150
continued next page
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Table 10 continued from previous page
Model
A
A
Rweak 4
Raneg 4
MMD, +
MMD 4
AM
0.0
1.93 10 (:: 0.32)
5.96 x 10 2.60)
0.024
0.008
AM
10k
0.0
1.11 x (4:
5.23 x 10
(#: 2.18)
0.063
0.012
AM
20k
0.0
9.15 (#2229)
4.71 10
2.13)
0.053
0.010
AM
20k
10
1.97 x 10 0.38)
5.13 x 10
(# 2.15)
0.010
0,010
AM
20k
1
1.64 10 0.34)
4.94 x 10'
(#: 2.20)
0.012
0.009
AM
20k
10-1 1
1.23 x 10 (1.027)
4.73 x 10 (1: 2.21)
0.023
0.006
AM
20k
10 2
1.05 (1
4.63 x 10* 2.08)
0.049
0.009
AM
20k
10 5
9.78
(+2288)
4.74 x 10 (# 2.19)
0.034
0.008
AM
100k
0.0
1.15 10 (#2228)
4.9 x 10' (+55)
0.006
0.006
AM
IM
1.26 10 (.32)
x 10' (1: 2.72)
0.006
0.008
AM
10M
0.0
1.54
10 (#
5.09 x 10 (# 2.69)
0.008
0.007
AM
100M
0.0
1.32
10 (# 0.31)
5.23 x 10 2.77)
0.013
0.009
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Published as a conference paper at ICLR 2026
F.1.2 ELASTICITY
x
0.100
0.075
0.050
0.025
0.005
0.004
0.003
0.002
0.001
Rusuc(xheery
Rx
Ru
(xhee
Rusell"")
Resuc(X
Figure 8: Fine-tuning towards boundary modification, comparing our approach with ECI, Samples for two random seeds shared across models. Top row: displacement fields. Bottom row: corre- sponding weak residual heatmaps.
Base AM
15
15
f
Base AM
Rweak (rel)
Reeak (rel)
AM
10
10
PBFM
K
0.2
0.4
0.6
0.8
0.1
0.2
0.3
0.4
0.5
MMDx
MMD
20
20
Restrong (rel)
Ratrong (rel)
15
15
10
5
0.2
0.4
0.6
0.8
0.1
0.2
0.3
0.4
0.5
MMD,
MMD
Figure 9: Elasticity dataset: scatter plots showing the relationship between PDE residuals (weak and strong) and distributional discrepancies (MMD,. MMD,) across all model variants and hyper- parameter configurations. Each point corresponds one configuration; lower values indicate better physics consistency or distributional fidelity.
Table 11: Elasticity dataset: full residual and distributional metrics for meth- ods and hyperparameter settings
Model
Rweak +
Ratrong +
MMD 1
MMD, +
FM
1.59 x 10 (#: 0.37)
1.83 x 10 (= 0.66)
0.238
0.049
PBEM
6.32 10 (#: 0.82)
4.22 x 10" (= 0.26)
0.918
0.544
Base AM
1k
6.78
10
(+ 1.09)
1.21
10 0.34)
0.325
0.194
Base AM
10k
4.62 x 10 (#: 0.85)
4.89 x 10 (= 1.29)
0.336
0.297
continued next page
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Table 11 continued from previous page
Model
A
X1
Rweak +
Rezonng 4
MMD, +
MMD. +
Base AM
50k
5.47 x 10 (# 0.85)
4.16 x 10 (= 0.87)
0.404
0.314
Basc AM
100k
6.18 x 10
(# 0.75)
4.0 x 10" (# 0.98)
0.396
0.288
Base AM
500k
6.68 x 10
(+: 0.79)
4.04 x 10 (# 0.76)
0.417
0.299
Base AM
IM
6.08 x 10" (#: 0.74)
3.76 x 10" (# 0.54)
0.446
0.288
Basc AM y
1k
6.99 x 10 (# 0.99)
1.06 x 10' (# 0.30)
0.323
0.180
Base yp
10k
4.58 10 (#: 0.70)
4.91 10 (# 1.31)
0.339
0.298
Base AM ip
50k
5.55 x 10 (#: 0.72)
4.58 x 10 (# 0.94)
0.359
0.290
Base AM y
100k
5.63 10 (# 0.82)
4.16 x 10 0.93)
0.375
0.288
Base AM &
500k
5.47 x 10 (+: 0.72)
3.92 x 10 (# 0.77)
0.408
0.296
Base AM 4
IM
6.39
10" (#: 0.86)
1.09 x 10 0.68)
0.469
0.286
AM
5k
5.57
10 (#: 0.60)
4.34 10 0.95)
0.510
0.248
AM
20k
Ik
8.33 x 10 (#: 1.34)
4.78 x 10 (# 1.15)
0.096
0.060
AM
30k
1k
8.44 10" (# 1.27)
4.48 10 (= 1.08)
0.091
0.060
AM
50k
1k
7.85 x 10 (# 1.17)
4.4 10 (# 1.08)
0.108
0.064
AM
100k
0.0
3.04 x 10 0.31)
5.06 x 10 (# 0.66)
0.144
0.253
AM
100k
6.78
10 (4: 1.00)
444 10 (# 1.02)
0.091
0.074
AM
500k
1k
6.15
x 10 0.77)
3.79 x 10 (= 0.87)
0.148
0.121
AM
1M
3.93
10 0.27)
3.49 10 0.48)
0.204
0.297
AM
1M
1k
5.14
x 10 (+ 0.42)
2.85 x 10 (= 0.58)
0.197
0.203
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Published as a conference paper at ICLR 2026
F.1.3
HELMHOLTZ
R-wesk (rel)
15
Rweak (rel)
AM
AM
FM
10
10
PBFM
0.075
0.100
0.125
0.150
0.175
MMD,
MMD
Rtroona (rel)
25
Restrong (rel)
25
20
20
15
1
15
I
10
10
0.075
0.100
0.125
0.150
0.175
(..000.0325003350
0et.0.5550003733 037500 .00.0.000...... 0...0...42.000450
MMDx
MMD
Figure 10: Helmholtz dataset: seatter plots showing the relationship between PDE residuals (weak and strong) and distributional discrepancies (MMD,., MMD,,) across all model variants and hyper- parameter configurations. Each point corresponds to one configuration; lower values indicate better physics consistency or distributional fidelity.
Table 12: Helmholtz dataset: full residual and distributional metrics for all meth-
ods and hyperparameter settings
Model
X Rwesk
Ratrong +
MMD, +
MMD. +
FM
1.5 10 (# 0.59)
2.55 10 0.55)
0.177
0.033
PBFM
8.33
10" 3.04)
x 10 0.33)
0.090
0.031
Base AM
5.64 x 10" 2.09)
1.59 x 10 (1 0.33)
0.134
0.039
Base AM
50k
4.9 x 10 (1: 1.85)
1.34 x 10 (1 0.32)
0.147
0.041
Base AM
100k
5.19 10 2.01)
1.27 10 0.32)
0.148
0.042
Base AM
IM
4.95 x 10 (t 1.81)
1.17 x 10' (+ 0.33)
0.144
0.042
Base AM
100M
5.01
10 2.00)
x 10 (1 0.32)
0.143
0.042
Base AM y
5k
5.46 10 (+ 1.94)
1.59 10' (+ 0.33)
0.124
0.039
Base AM yp
50k
5.35 x 10 (1)1
1,35 x 10 (# 0.34)
0.133
0.045
Base AM 1
100k
5.02 x 10 (#2.77)
1.32 x 10 (# 0.33)
0.126
0.043
Base AM + Pp
1M
4.99 x 10 (.12)
1.16 x 10 (# 0.33)
0.125
0.046
Base AM Sp
100M
5.41 x 10 (+ 2.20)
1.17 x 10 (t 0.34)
0.127
0.044
AM
5k
I
5.32
10 (#: 1.49)
1.48 x 10 0.29)
0.081
0.035
0.0
4.71 x 10 (4: 1.26)
1.35 x 10 (t 0.28)
0.069
0.037
AM
100k
0.0
4.3 (4
1.14 X 10 (+
0.069
0.037
AM
100k
0p1
4.49
(1
x 10 (.29)
0.069
0.040
AM
100k
1
4.42
142x100 10 1.33))
1.21 X 10 (#
0.068
0.041
AM
100k
1k
8.36
x 10 (.66))
1.16 X 10 (.25)
0.151
0.032
AM
500k
0.0
4.56 x 10 (.47)
1.09 X 10 (1 0.31)
0.066
0.046
AM
IM
0.0
4.43 x 10 (1 1.43)
1.1 X 10' ( 0.30)
0.064
0.046
IM
1k
5.83
x 10 (+ 1.63)
1.1 X 10 (= 0.27)
0.106
0.032
AM
100M
0.0
4.43 x 10 (+ 1.43)
1.1 X 10' (= 0.30)
0.065
0.045
continued on next page
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Table 12 continued from previous page
Model
A
Rweak
Rawona 4
MMD,
+
MMD. 4
AM
100M
4.32 x 100 (# 1.43) 1.05 x 10 (# 0.30)
0.062
0.042
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F.2
STOKES LID-DRIVEN CAVITY
100
100
Base AM
Rweak (rel)
80
Rweak (rel)
80
Base AM
60
60
AM
FM
40
40
PBPM
I I
I
20
20
0
0
0.2
0.4
0.6
0.8
1.0
0.3
MMDx
MMD
6
I
Ratrong (rel)
Ratrong (rel)
4
2
0
0.2
0.4
0.6
0.8
1.0
0.05
0.10
0.15
0.20
0.25
0.30
0.35
MMDx
MMD,
Figure 11: Stokes dataset: scatter plots showing the relationship between PDE residuals (weak and strong) and distributional discrepancies (MMD,, .MDD.) across all model variants and hyperpa- rameter configurations. Each point corresponds to a single configuration; lower values indicate better physics consistency distributional fidelity. PBFM is omitted in the strong-residual panels because its training did not converge to physically meaningful samples (yielding strong residuals several orders of magnitude larger than the other methods.
Table 13: Stokes dataset: full residual and distributional metrics for all methods and hyperparameter settings.
Model
d
Rowak
Ratrong
MMD,
MMD,
FM
3.05 x 102 (# 3.16)
1.81 x 10 (#: 0.44)
0.224
0.049
PBEM
9.94 x 10" (+ 2.17)
1.15 x 10
(#: 0.05)
0.227
0.071
Base AM
5k
1.42 x 10 (#: 0.28)
1.47 x 10
(#: 0.17)
0.484
0.212
Base AM
10k
1.16 x 102 (# 0.27)
1.43 10
(# 0.13)
0.500
0.233
Basc AM
100k
6.88 x 10 (t 2.47)
1.07 x 10
(#: 0.14)
0.561
0.266
Basc AM
1M
5.3 x 10 (#: 2.23)
9.94 10
(# 1.59)
0.611
0.267
Base AM
100M
x 10 (1 2.16)
9.04 10 (# 1.77)
0.749
0.271
Basc AM +
5k
1.22 x 10 (t 0.22)
1.37 x 10 (+: 0.12)
0.515
0.236
Base AM
10k
9.0 x 10 (1 1.55)
1.3 x (1 0.12)
0.532
0.256
Base AM + yp
100k
x 10 2.28)
1.15 10 (1 0.13)
0.514
0,250
Base AM +
1M
8.55 x 10 (t 2.52)
1.16 x 10 (+ 0.13)
0.535
0.252
Base AM 4p
100M
4.64 10 2.04)
7.89 10-11-101 (+ 1.59)
0.825
0.267
AM
10k
0.0
1.41 10a 0.93)
1.16 10 (+ 0.20)
0.426
0.087
AM
10k
1
1.98 x 10 0.77)
1.3 10 0.24)
0.452
0.095
AM
50k
0.0
9.14 10 4.8))
1.05 10 (+ 0.20)
0.514
0.164
AM
100k
0.0
7.0 10th 3.51)
1.07 x 10 0.20)
0.570
0,219
AM
100k
1k
3.43 x 10 1.80)
4.87 10 (# 1.29)
0.186
0.107
500k
0.0
6.26 x 10 3.15)
1.07 10 (+ 0.21)
0.605
0.235
AM
IM
0.0
5.8 x 10 3.28)
1.03 10 (+: 0.20)
0.632
0.229
IM
1k
x 10 2.73)
x 10 0.30)
0.411
0.081
AM
100M
0.0
4.72 x 10 (# 2.84)
1.11 X 10 (+ 0.15)
0.882
0.266
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Table 13 continued from previous page
Model
A
A
Rweak
Rawogg
MMD,
MMD
AM
100M
100k
3.67 x 10 2.47)
1.35 x 10 (#: 0.34)
0.515
0.112
AM
100M
10k
1.45 x 10' (# 0.34
1.64 10"
(+ 0.31)
0.678
0.131
AM
100M
1
x 10 3.43)
1.02 x 10"
(+ 0.16)
0.885
0.260
AM
100M
1k
5.27 x 10 2.36)
1.07 x 10 (# 0.12)
1.036
0.354
F.2.1
EFFECT OF FORCING MISSPECIFICATION
In the Stokes did-eriven cavity experiment, the base Flow Matching (FM) model is trained data generated with a nonzero Kolmogorov-type forcing term with amplitude Fo 2.0. In the main experiment reported in the paper, however, fine-tuning is performed under the assumption of forcing (Po 0.0). This represents a severe form of model-physics misspecification: the fine- tuning objective assumes a qualitatively different flow regime from the data on which the FM model was trained.
(a) Fo 2.0
(b) Po 1.5
(c) Po 1.0
(d) Fo 0.5
(c) Po 0.0
Figure 12: Representative stationary velocity fields for the same viscosity but different forcing am- plitudes Fo. With Fo = 0, the flow is driven solely by the moving lid and exhibits single dominant recirculating vortex. As the forcing strength increases, additional sub-vortices emerge and the flow develops progressively more complex interior structures.
Figure 12 illustrates this mismatch. Samples with Fú = 2.0 exhibit complex flow structure with multiple vortices driven by the interior forcing. In contrast, the flow for Fg 0.0 contains essen- tially a single large recirculating vortex induced only by the moving lid. These differences reflect fundamental change in the underlying PDE solution manifold rather than minor perturbation. Fine-tuning the FM model to match such a drastically different flow regime therefore constitutes an inherently challenging, strongly out-of-distribution adaptation problem.
To study this effect more systematically, we perform additional fine-tuning runs using different assumed forcing amplitudes Fo E {0.00 0.5, 1.0, 1.5, 2.0), while always starting from the same FM model trained on data with Fo 2.0. All other hyperparameters are kept fixed (Ag Aa- 100k,
A 0.0). For each Fo. compute the weak PDE residuals after fine-tuning and report them relative to the mean weak residual of ground-truee truth data generated with the same forcing amplitude Fo. Because both pre-training of the inverse predictor and the fine-tuning itself use the same assumed forcing Fo. we can also report baseline values for the FM model by applying the corresponding inverse predictor to its samples. Figure 13 presents the resulting residuals.
Across all levels of assumed forcing, fine- tu-ing consistently reduces the weak residuals by at least an order of magnitude compared to the corresponding FM baseline. As expected, the remaining residuals strongly depend on the degree of misspecification: when Fo is close the true value (2.0), the relative residuals after tuning approach 1.0, indicating consistency with the ground-truth distribution. Under severe mismatch (e.g. Fo 0.0), the residuals remain higher despite signifi- cant improvement relative to the FM baseline, which is consistent with the demand for structurally significant modification.
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200
100
Rweek (rel)
5
I
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
FFo - Fol
Figure 13: Weak residuals of the base FM model (red) and the fine-tuned model (black) as a function of the assumed forcing amplitude. Reduced misspecification leads to substantially lower post-tuning residuals, approaching the ground-truth residual level when the assumed forcing matches the true physical value.
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F.3 NON-CURATER SAMPLES
F.3.1 DARCY
Base FM model
(b) fine-tuned with \ 1.0
(c) Fine-tuned with '5
Figure 14: Darcy flow: non-curated samples of pressure distributions (left columns) and recovered permeability fields (right columns). Each row was generated using the same initial noise across the three models. Color scales are normalized per row for the pressure distributions. For the base model, permeabilities obtained with the pre-trained inverse predictor.
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F.3.2 ELASTICITY
(a) Base FM model
(b) Fine-tuned with model
Figure 15: Non-curated samples from the elasticity experiment, where fine-tuning has to scale down the lower boundary.
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F.3.3 HELMHOLTZ
(a) Base FM model (damped)
(b) Fine-tuned (lossless)
Figure 16: Non-curated Helmholtz samples. In both panels, each shows: (i) the real part of the generated field, (ii) its imaginary part, and (iii) the recovered wave-speed parameter c.
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Published as a conference paper at ICLR 2026
F.3.4 STOKES
Base FM model (forced)
(b) Fine-tuned (no forcing)
Figure 17: Non-curated Stokes samples. Each row displays: (i) the viscosity field v, (ii) the velocity magnitude ||w]], and (iii) streamplot of the velocity (un, Uy).
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Published as a conference paper at ICLR 2026
F.3.5 GUIDANCE
Figure 18: Guided samples with increasing number of given observations, specifically [25, 50, 100, 200, 300, 400, 500, 750, 1000]. For each number of conditioning points, we generate two samples from independent noise and condition on the same sparse samples, indicated as white markers in the right column. As expected, with more points both & and a become more constrained and less diverse.
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Published as a conference paper at ICLR 2026
F.3.6 NATURAL IMAGES: RECOLORING
(a) Base LFM model
(b) Vanilla Adjoint Matching
(c) Joint Adjoint Matching
Figure 19: Non-curated independent samples from LFM model conditioned on class "macaw" and using guidance scale 4.0. Models were fine-tuned to maximize PickScore using the prompt "close- up pop art of macaw parrot".
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Published as a conference paper at ICLR 2026
(a) Base LFM model
(b) Vanilla Adjoint Matching
(c) Joint Adjoint Matching
Figure 20: Non-curated independent samples from LFM model conditioned on class "boathouse" and using guidance scale 4.0. Models were fine-tuned to maximize PickScore using the prompt "boathouse with green and purple curtains of northern lights." Our joint model is able to generate the colors demanded in the prompt while retaining diversity in the generated boathouses.
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Reviews:
[{"review_id": "150xxQEo77", "reviewer": "Reviewer_Mbyw", "summary": "The paper presents a post-training scheme that takes a pretrained flow-matching generator and tilts its distribution toward PDE-consistent samples by minimizing weak-form PDE residuals—so you can enforce physics (and boundary conditions) without retraining from scratch or having paired (state, parameter) data. Fine-tuning is posed as Adjoint Matching (a memoryless stochastic optimal control) with a small theoretical extension that scales the noise schedule for stability.", "questions": "Since physics is enforced via weak-form residual penalties (not hard projections), could you report post–fine-tuning feasibility diagnostics—e.g., distributions of PDE residual norms, boundary-condition violations, and conservation drift?\n\nCan you provide sensitivity curves (accuracy and residuals) versus the hyperparameters across datasets, along with any principled guidance or conditions that ensure stability without ad-hoc tuning? \n\nBecause dense weak-residual evaluation is costly, you use compact test functions and patch-based subsampling. Could you quantify coverage (e.g., how many test centers are required to achieve a target residual error) and explore adaptive sampling that targets high-residual regions?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "-The key selling point is the enforcement of PDEs via weak-form residuals on a pretrained flow-matching model—no paired data or full retraining while keeping the base model’s inference cost.\n\n-The model jointly evolves state and latent parameters with an inverse predictor, enabling guided sampling from sparse parameter observations and adaptation under model misspecification.", "weaknesses": "The proposed method is practical for post-training physics enforcement for flow matching (no paired data or full retraining). It is useful and timely, but quite incremental rather than foundational.\n\nPhysics is imposed via a weak-form residual penalty added to the flow-matching objective; it aligns the denoiser with PDE residuals but does not guarantee exact constraint satisfaction.\n\nThe method relies on several heuristics and hyperparameters, such as scaled “memoryless” noise with factor κ, time-grid tilting (q = 0.9), and computing loss on only a subset of late steps (K_last, K). These improve stability but introduce tuning sensitivity without a comprehensive robustness study.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "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": "", "extra_scores": {}}, {"review_id": "3t7QZxiKLP", "reviewer": "Reviewer_ZYUG", "summary": "The paper introduces a physics-constrained fine-tuning framework for pretrained flow-matching generative models, enabling them to satisfy PDE-based physical laws and jointly infer latent parameters without retraining from scratch. It casts the physics-constrained fine-tuning as  Adjoint-Matching loss for distribution-level correction and a lightweight architectural extension with residual heads for joint state–parameter evolution. Overall, it positions itself as a general and data-efficient bridge between physics-informed learning and modern generative modeling.", "questions": "Q1: How does the proposed fine-tuning framework scale to nonlinearity where PDE constraints become non-analytic or high-dimensional? Are there stability or performance guarantees in such regimes?\n\nQ2: Could the authors clarify which aspects of their method represent core methodological contributions beyond the existing Adjoint Matching framework (Domingo-Enrich et al., 2025)?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "**S1.** The paper recasts physics-based simulation as an adjoint-matching control framework, elegantly linking preference-aligned generative fine-tuning with physics-constrained inference. This bridges simulation-augmented modeling and stochastic optimal control, enabling physically consistent generative trajectories.\n\n**S2.** The method’s joint treatment of state and latent parameters allows simultaneous forward generation and inverse recovery within a unified flow-matching model.\n\n**S3.** The experimental evaluation is broad and convincing, demonstrating consistent improvements in physics residuals and inverse reconstruction accuracy across multiple PDE benchmarks, while maintaining generative fidelity and efficiency.", "weaknesses": "**W1**: A core conceptual weakness of the paper is that it does not truly model the joint state–parameter distribution $(x, α)$. The latent variable $α$ is introduced post hoc through a frozen inverse predictor $\\phi(x_1)$, which breaks end-to-end coupling between physical states and governing parameters. I would be interested know what tradeoffs do this approach have. An ablation where the base model jointly learns $(x, α)$ during pretraining would help clarify the effectiveness of the two-stage setup or if a single joint flow $v_t(x, α)$ could achieve stronger physical coherence and lower residuals.\n\n\n**W2:** The mathematical notation is dense and inconsistent, making the exposition difficult to follow even for technically skilled readers. Symbols like ($v_t$), ($b_t$), and ($u_t$) are overloaded across base, fine-tuned, and control flows, while stochastic and deterministic forms are interleaved without clear separation. Algorithm 1 is not self consistent, missing how for e.g. $\\phi$ is used. As a result, the formalism obscures theoretical contributions and could benefit from a clearer hierarchy of variables (e.g., consistently distinguishing state vs. parameter flows) and unified notation across sections.\n\n**W3:** The theoretical novelty of the paper is incremental over the original Adjoint Matching framework (Domingo-Enrich et al., 2025). The core stochastic optimal control formulation, adjoint dynamics, and lean-adjoint optimization are directly inherited, with the main contribution being their adaptation to PDE-constrained fine-tuning. While this is a valuable and well-motivated extension, it constitutes more of an application-level adaptation than a fundamentally new methodological advance.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "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": "", "extra_scores": {}}, {"review_id": "zOIoPgVHjN", "reviewer": "Reviewer_Pud7", "summary": "This paper proposes a post-training framework that fine-tunes flow-matching generators using randomized weak-form PDE residuals and a joint latent-parameter pathway, so the model produces physics-consistent fields while simultaneously inferring hidden coefficients; experiments on canonical PDE tasks indicate reduced residuals with limited impact on sample diversity.\n\nContributions.\n1) Post-training physics alignment: turns an already trained flow-matching model into a physics-respecting generator by minimizing weak-form residuals with compact test functions, avoiding unstable high-order derivatives and limiting drift from the base distribution.\n2) Joint state–parameter generation: augments the generator with a learned evolution for latent physical parameters and an inverse predictor, and fine-tunes both under an adjoint-matching objective (with a scaled memoryless schedule) to couple solutions and parameters.\n3) Practical control and coverage: demonstrates denoising, sparse-observation guidance, and boundary-condition adaptation, and exposes simple knobs to trade off constraint strength versus fidelity/diversity, with lightweight fine-tuning overhead.", "questions": "1) Diversity vs. PDE correctness. For well-posed forward/inverse PDEs, please justify when output diversity is desirable; otherwise, replace or augment SSIM-based diversity with task metrics (solution error L2/H1, weak/strong residual distributions, boundary-violation rates) and, for partial-observation settings, include posterior calibration (coverage vs. nominal).\n\n2) Scope of test problems. Add at least one oscillatory elliptic case (Poisson/Helmholtz) and one basic incompressible flow (e.g., lid-driven cavity or cylinder shedding); if new runs are infeasible, provide higher-resolution or 3D variants or a brief scaling analysis (compute, stability bottlenecks).\n\n3) Baselines and recovery metrics. Include matched-compute head-to-head with (i) training-time physics-regularized flow matching, (ii) inference-time projection/ECI, and (iii) a classical PDE-constrained inversion baseline. Report solution error, boundary violations, weak/strong residuals, latent-parameter MAE/RMSE, and wall-clock.\n\n4) Ablations for method choices. Provide a small ablation comparing weak vs. strong residuals (stability, final residuals) and sensitivity to test-function sampling; show how the scaled memoryless parameter kappa affects stability, residual reduction, and drift from the base distribution.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1) Proposes a post-training route to impose physics on pretrained flow-matching models via weak-form PDE residuals, coupled with joint latent-parameter evolution for inverse problems without paired labels; also introduces a scaled memoryless noise schedule within adjoint matching.\n\n2) Grounds the method in adjoint matching and implements randomized local test functions for stable weak residuals; experiments span Darcy denoising, sparse-observation guidance, linear-elasticity boundary adaptation, and a small natural-image recoloring case, with ablations showing a residual–diversity trade-off.\n\n3) Clearly states goals and contributions, provides a pipeline diagram, derives the weak forms and test-function design, includes a full training algorithm and detailed dataset/backbone specs, and offers a reproducibility statement.", "weaknesses": "1) Diversity objective may be misaligned for PDE solvers. For well-posed forward/inverse PDEs the target is a single solution; promoting output “diversity” is not desirable, and when partial observations make the task ill-posed, diversity stems from the problem, not the pipeline. The paper treats diversity as a knob/metric (SSIM-based) and studies its trade-off against residuals (Fig.\\ 3), which can conflict with PDE goals.\n\n2) Test problems are not comprehensive. Evaluations focus on Darcy denoising, sparse-obs guidance, and a linear-elasticity boundary change, plus a small image recoloring demo; there is no coverage of more challenging PDEs such as Poisson, Navier–Stokes, or Helmholtz, nor larger-scale or multi-physics settings.\n\n3) Limited baselines and quantitative recovery metrics. Beyond an ECI comparison in the elasticity case, there is no systematic head-to-head with alternative physics-constrained generative methods, and the paper provides little quantitative evaluation of latent-parameter recovery accuracy or real-data tests (most results are residual reductions and visuals).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "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": "", "extra_scores": {}}, {"review_id": "voHngDO1tb", "reviewer": "Reviewer_n7dD", "summary": "This paper introduces a physics-consrtained post-training scheme for pre-trained flow-matching generators that enforces PDE consistency and simultaneously infers latent physical parameters for inverse problems without paired (state, parameter) supervision. The key idea is to treat weak-form PDE residuals as a reward and fine-tune the generator via Adjoint Matching. Experiments on Darcy flow, linear elasticity and natural images demonstrate PDE residual reduction and parameter recovery.", "questions": "Please refer to weaknesses section.", "limitations": "", "rating": null, "confidence": 4, "soundness": 1, "presentation": 3, "contribution": 1, "strengths": "## Originality\n- Novel problem formulation. Enforcing parameter-dependent PDE constraints without paired (parameter, solution) training data.\n- Creative architecture design. Joint state-parameter evolution with surrogate base flows constructed via inverse predictor.\n- Technical contribution. Scaled memoryless noise schedule (Lemma 1) provides useful numerical stabilization.\n- Weak-form residual approach.Stochastic sampling of Wendland-wavelet test functions is practical and avoids instabilities of strong-form residuals\n\n## Quality\n- solid theoretical grounding via adjoint matching framework.\n- thoughtful multi-faced regularization with clear ablations.\n- diverse experimental scenarios covering denoising, sparse conditioning and boundary misspecification\n- extensive reproducibility details \n\n## Clarity\nGood visual explanation, well-motivated problem setup\n\n## Significance\nPost-training paradigm is more practical than training from scratch. Proof-of-concept establishes feasibility of joint parameter-solution generation post-training, potentially inspiring future work.", "weaknesses": "## Unacceptable Absence of Quantitive Results\nThe paper is almost entirely based on qualitative visualizations with virtually no quantitative evaluation. Only Figure 3 quantifies how the hyperparameters mediate the trade-off between staying close to the base model and reducing the (weak) PDE residual. Everyting else is visualizations of cherry picked samples. No quantitive parameter recovery metrics despite solving \"inverse problems\". No numerical residual comparisons with other methods despite claiming constraint enforcement. No baseline comparison tables of even one numerical metric despite comparing with ECI. \nThe absence of quantitive results makes it impossible to assess\n- Whether the method actually works reliably?\n- How it compares to alternatives (I am not talking about a comprehensive benchmarking agains other SOTA methods)\n- When it succeeds vs fails\n- what design choice actually matter.\n\nThe authors are encouraged to conduct thorough quantitative experiments for future submissions of this work.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)