Papers
Machine learning for physics, as it is published.
New work from arXiv on CFD surrogates, neural operators and physics-informed learning, refreshed every hour. Search it, filter by topic, or start with the papers we keep coming back to.
Papers
New work from arXiv on CFD surrogates, neural operators and physics-informed learning, refreshed every hour. Search it, filter by topic, or start with the papers we keep coming back to.
2,359 papers in Neural operators · newest first
cs.LG
Neural operators provide efficient surrogates for computational fluid dynamics (CFD), but most are designed for static geometries and are not equipped to model unsteady flows induced by continuously moving boundaries. We study dynamic-boundary 4D CFD prediction for articulated moving geometries and introduce RoboCFD-4D, a temporally resolved robotic CFD benchmark spanning diverse morphologies and flow conditions. To address the computational and autoregressive challenges of this setting, we propose DynaSolver, a neural operator that explicitly conditions on boundary evolution and causal flow history. DynaSolver employs an efficient physical-token formulation that amortizes point-to-token projection across the operator stack, substantially reducing repeated point-space spatio-temporal computation while retaining accurate point-level reconstruction. We further introduce Replay-Augmented Parallel Training, which exposes the model to autoregressively generated histories while preserving parallel dense supervision, narrowing the gap between ground-truth-conditioned training and autoregressive rollout. Experiments on RoboCFD-4D show that DynaSolver achieves the best full-rollout accuracy among a broad range of learned CFD surrogates, with its advantage increasing as rollout proceeds, indicating substantially reduced autoregressive error accumulation. Project page: https://made-gpt.github.io/DynaSolver-ProjectPage.
physics.comp-ph
Machine-learning surrogates are increasingly proposed to replace phase-field fracture simulations in many-query tasks such as uncertainty quantification, inverse analysis and design. Little controlled evidence shows which architectures are reliable, for which fracture problems and how far beyond their training data. We introduce Fracture-Bench, an open benchmark of two phase-field datasets with fixed splits, common test metrics and a description of each dataset's physical regime. Dataset~1 contains 1,000 AT2 simulations of multiple cracks growing from a notched hole in a hyperelastic plate under random non-uniform biaxial boundary displacements (44 load steps, grid). Dataset~2 contains 4,000 PF-CZM simulations of a pre-cracked plate with two inclusions, each ten times stiffer or more compliant than the matrix ( grid). We train a Fourier neural operator (FNO), DeepONet, Latent DeepONet, a UNet and a conditional diffusion model on identical splits and compare one test metric per dataset, training time, inference time and model size. On Dataset~1, FNO and Latent DeepONet attain mean test errors of order , about ten times lower than UNet and DeepONet. On Dataset~2, the diffusion model is most accurate, with about one sixth of the UNet error, but requires \SI{706}{s} per prediction versus \SI{6}{ms} for UNet. For all architectures, most of the error occurs in the final load steps, after crack localization. Dataset~2 contains initial cracks less than one grid cell apart that produce qualitatively different paths. Architecture rankings therefore depend on the task, and errors averaged over densely sampled inputs reveal little about near-bifurcation cases. Data, codes, splits, regime descriptors and generalization splits are public, with a protocol for adding under-represented regimes such as mixed-mode, three-dimensional and experimentally paired fracture.
cs.LG
Spherical-harmonic neural operators provide an efficient representation for global forecasting, but their discretized pointwise nonlinearities can break rotational equivariance through aliasing. Clebsch-Gordan (CG) tensor products provide exactly equivariant interactions between spherical-harmonic representations, but their computational cost grows rapidly with spectral bandwidth and has limited their use at forecasting scale. We introduce LCG-SFNO, which combines low-rank CG interactions within a controlled low-degree band with coefficient-space residual propagation and an exactly projected quadratic nonlinearity, preserving SO(3) equivariance throughout the learned operator. To make the CG interaction practical, we further develop SSHCG, a sparse GPU implementation with a factorized alternative for wider coupled bands. On spherical shallow-water dynamics and the ERA5-based WeatherBench benchmark, LCG-SFNO reduces rotational error to numerical precision (more than three orders of magnitude below SFNO on shallow water and five orders of magnitude below it on WeatherBench), while matching SFNO on shallow-water forecasting and substantially improving forecast skill relative to SFNO on WeatherBench. Matched ablations show that CG coupling accelerates convergence, while SSHCG keeps the coupling to about 10 percent of a training step at the operating bandwidth, making exact equivariant coupling practical at forecasting scale.
cs.LG
Physically based two-dimensional hydraulic models are too slow for real-time inundation forecasting and large ensembles. We present a time-dependent geometry-informed neural operator (GINO) surrogate of LISFLOOD-FP for the municipality of Forlì (Italy), severely affected by the May 2023 Emilia-Romagna floods. The surrogate is designed for an operational setting in which only river stage gauges and rain gauges are available. A reproducible pipeline turns public regional observations into a physically based training set: each river inflow is driven by the first stage gauge downstream of it through a calibrated water-surface boundary condition, and rain enters as SCS-CN effective rainfall. The surrogate is an autoregressive GINO trained on 216 simulations of 24 observed events with scaled forcing. It predicts volume-conserving polygon-mean depths on an exposure-based Voronoi mesh of 659 nodes, with a 20 min step over 72 h. The surrogate is tested on 38 independent observed events that it never saw during training. On these events it reaches a critical success index (CSI) of 0.934 at 0.05m and 0.844 at 0.3 m, a depth RMSE of 0.008m and an of 0.984 for maximum-depth maps. On one NVIDIA A10G GPU a 72-h forecast takes 1.7 s instead of 55 s with LISFLOOD-FP, and a 50-member ensemble, computed in a single batched rollout, takes 6.2 s instead of about 46 min, which makes probabilistic flood forecasting practical. These results show that neural-operator surrogates can deliver accurate pluvial-fluvial flood forecasts driven by river stage for operational early warning.
cs.LG
Geometry-aware neural operators face substantial computational and memory demands when learning across complex physical domains. We introduce Architecture-Aware Mixed Precision (AA-MP), a replica-exchange training strategy that allocates numerical precision according to replica roles and architecture-specific operations. By retaining FP32 for refinement while selectively employing BF16 for exploration, AA-MP preserves predictive accuracy while improving training efficiency. Across fluid dynamics and three-dimensional structural mechanics benchmarks, AA-MP achieves comparable accuracy to full-precision training over an ensemble of runs, while reducing hot-replica peak GPU memory by 46.9% and training time by up to 25.8%. It also outperforms naive Automatic Mixed Precision (AMP) in memory efficiency and computational cost, with increasing memory advantages at larger model and geometry scales. Ablation studies show that retaining FP32 in the cold-replica is important for predictive accuracy, while additional FP32 operations in the exploratory replica provide limited benefits relative to their computational cost. Finally, conformal prediction for uncertainty calibration shows that AA-MP achieves near-nominal coverage with narrower prediction intervals than the alternatives. These results advance the scalability of geometry-aware scientific machine learning by reducing computational costs without compromising predictive accuracy or uncertainty quantification.
cs.LG
Physics-driven design seeks high-performing geometries, which need not conform to a learned shape category. Parametric and latent-space approaches facilitate surrogate-based optimization, but restrict deformations to prescribed shape families or learned decoders. We introduce the Mesh-to-Shell Neural Operator (M2S-NO) as the mesh-to-shell component of a category-agnostic inverse shape optimization framework. We separate the explicit mesh used for design from the volumetric representation used for physics prediction. M2S-NO maps oriented surface samples to a surface-centered shell representation, such as TruncatedSDF or HeatSDF. A separate problem-specific PDE neural operator (PDE-NO) maps this representation to physical fields. Objective gradients update mesh vertices through both operators, without a latent shape decoder. Preconditioning and remeshing support free-form deformation, while fixed interfaces are controlled directly on the mesh. Optional shell-space proposals enable topology changes. We evaluate geometric matching, jet-engine brackets, and neutron moderators, with independent FEM and OpenMC evaluation. Relative to the best training designs, M2S-NO reduces maximum bracket displacement by 14.3% and increases neutron-sensor current by 13.8%. Optimization also succeeds from an out-of-distribution bracket initialization, demonstrating the framework's geometric flexibility.
cs.CE
Predicting mechanical responses across complex geometries and varying loading conditions is central to simulation-driven engineering, yet capturing the influence of geometric boundaries remains challenging for data-driven models. Here we propose a Load- and Geometry-adaptive Deep Operator Network (LG-DON) that integrates global geometric representations with local, boundary-aware information to predict structural responses. By distinguishing distances to different boundary categories rather than relying on the nearest boundary alone, the framework captures spatially varying geometric influences that conventional distance-based representations may overlook. Across benchmark problems involving diverse non-convex geometries, LG-DON reduces the mean relative L2 error from 12.17% for a modified MIONet baseline to 7.41%. Controlled ablation studies further demonstrate the advantages of category-resolved boundary distances and show that combining global and local geometric representations outperforms either representation alone in 15 of 18 evaluated configurations. These findings highlight the value of integrating complementary geometric information for accurate operator learning across complex structural domains.
cs.LG
Three-dimensional magnetohydrodynamic (MHD) turbulence, the flow of astrophysical and fusion plasmas, is costly to simulate. We present, to our knowledge, the first learned surrogates of three-dimensional resistive MHD turbulence, and compare three standard backbones in that role: a convolutional UNet, a shifted-window transformer and a Fourier neural operator. They are trained as autoregressive emulators of decaying MHD turbulence at a resolution of and Reynolds numbers of 1000, 3000 and 6000, on 500 simulations per Reynolds number, each with one two-stage procedure, single steps and then short rollouts, and stopped by one rule. Over a window of about one large-eddy turnover time, the UNet and the transformer reach relative velocity errors of 0.018 to 0.053 and keep the current sheets, the enstrophy, the current density and the energy spectra of the simulation; which of the two is the more accurate depends on the Reynolds number, the UNet leading at 3000 and 6000 and the transformer at 1000. Applied with the same backbones and training procedure to two public benchmarks, the models reach the lowest error reported to date on the isotropic turbulence set of FactFormer, 39% below the best published model. On the compressible MHD set of The Well, their rollout errors are a third to a half below those of the best of The Well's baselines. They also match a 1.3 B-parameter pretrained foundation model in its rollout windows, with 20-62 times fewer parameters. The code, the configuration of every model, the evaluation scripts and the data generator with the seed of every simulation will be released upon publication.
cs.CE
In multiscale systems, nonlocal interactions among spatially distributed constituents can unlock collective, potentially emergent behavior irreducible to individual constituent responses. Beyond characterization, nonlocality in multiscale systems is increasingly viewed as a design element across fracture mechanics, wave physics, and disordered systems. Existing nonlocality theories provide rigorous descriptions and mechanistic insight, but often require prescribed analytic forms, range, symmetries, or periodicity. Machine learning offers a data-driven route to learn interactions, yet principled approaches that tie the network design to physical theory remain nascent. Here we present domain-decomposition attention neural operator (DANO), which learns configuration-dependent interactions in heterogeneous systems governed by linear field equations from field data alone. Through a case study in large-scale optical metasurfaces, we demonstrate DANO's full-field prediction under geometric heterogeneity and across drive conditions, small-to-large domain transfer, and multifunctional inverse design, with learned interactions validated through physics-based interrogation. As an AI-for-Science tool, it supports interaction discovery and scalable, customizable design of nonlocal multiscale systems, including virtual-reality optics.
math.OC
This paper focuses on the use of learning nonlinear operators via Deep Operator Networks (DeepONets) for approximating the solutions of optimal control problems. We study two operator learning strategies, namely, indirect and direct, both based on DeepONets, for parametric families of optimal control problems where the same problem must be solved many times for different initial data, different parameters, different cost function weights, and different bounds defining the set of the admissible controls. In the indirect strategy two DeepONets approximate the solution operators of the state and adjoint systems, and their predictions are used inside iterative projected gradient algorithms for finding the optimal control. In the direct strategy, a single DeepONet approximates the operator that maps directly to the optimal control generated by a reference solver; consequently, a new control is obtained through a single evaluation of the network. In order to analyze the computational cost, the efficiency, the speed and the accuracy of the learning strategies, we consider an optimal control problem related to an ordinary differential system that describes the temporal evolution of the glioblastoma growth subject to therapeutic interventions. For this specific optimal control problem we propose a discretize-then-optimize strategy, based on an implicit-explicit finite difference method, and then, we develop several numerical experiments in order to compare the two DeepONet learning strategies for this problem. The proposed operator learning DeepONet strategies can be adapted to approximate large classes of optimal control problems related to diverse differential systems.
math.NA
We introduce NEOPIC, a framework for learning the field operator in particle-based methods applied to kinetic plasma simulations. Its key novelty is a neural operator that acts directly on the empirical measure of the particles, rather than on a grid, and is evaluated at the particle positions. No mesh or tree is needed and, unlike grid-based neural operators, no deposition onto a mesh takes place, so the particle noise that dominates the inputs of particle simulations is not aliased. Since it replaces only the field solve, it serves as a plug-and-play field solver for particle-based methods: it can be trained on data from particle-in-cell, particle-in-Fourier or tree codes, and combined with any time integrator, including implicit schemes. We prove that the discrete evaluation coincides with the continuous operator at any number of particles (representation equivalence) and derive an error analysis that separates sampling, learning and time-stepping errors: up to the learning error, NEOPIC has the error structure of standard Monte Carlo particle methods. We show that the particle noise in the training data acts as an implicit regularizer, which gives guidance on the number of training particles in relation to the particle numbers used at inference. A complexity analysis shows that NEOPIC is an method whose constants are set by the architecture alone, independently of any mesh or tree. Numerical experiments on electrostatic benchmarks in two and three dimensions, including Landau damping, two-stream and bump-on-tail instabilities, a cyclotron mini-app and a Penning trap, confirm the theoretical findings. A single model trained on several mini-apps generated by different reference methods captures all of them and remains stable up to twice the training time window, which points towards foundation models for fields in particle-based methods.
math-ph
High-fidelity CFD is essential for coupled hypersonic aerodynamic and aerothermal phenomena, but its cost precludes multi-objective design optimization. We develop three DeepONet surrogates for two hypersonic configurations, unified by one device: the trunk receives coordinates in which the varying feature is held fixed. The first is a prescribed family of 37 spherically blunted tangent ogives, varied through ogive radius of curvature to span nose-tip ablation forebody change; no ablation is computed. A body-fitted canonical transformation maps every domain onto a common (s, eta) space, retaining physical coordinates for forces and heat transfer. A branch network predicts reduced coefficients and a Transolver trunk with Physics-Attention learns the spatial basis; a two-step strategy optimizes both jointly, orthonormalizes the basis by QR factorization, and trains a second-stage branch for unseen geometries. Surrogates cover density, pressure, velocity, temperatures, conductivities and species mass fractions. In leave-one-out tests, drag, mean and peak wall heat flux are recovered to 1.16%, 1.62% and 5.29%; a sweep of 1001 geometries gives a 200-design Pareto front trading 6.68% in drag for 4.42% in peak heating (a diagnostic, not an objective). The second is the 3D AEDC waverider at 29 angles of attack: static mesh, moving leading-edge shock. For surface quantities, the same construction predicts integrated heat load within 0.22% and force coefficients to errors below 2.6x10^-5; Pareto fronts identify angles of attack trading drag, lift and heating. For the volume field, a Cartesian trunk smears the shock; a frozen ShockNet supplies a shock-aligned coordinate and a hotspot-aware mixture of experts corrects the residual, recovering density to 12.28% relative L2 error at a held-out incidence (5.08x10^7 nodes). The 1001-design sweep takes about three minutes on a single CPU.
math.NA
We propose a semigroup-aware structured kernel neural operator for source-driven evolution PDEs. Unlike generic time-conditioned kernel models, the proposed formulation represents temporal dependence through a latent exponential propagator \(e^{tA_θ}\), where \(A_θ\) is a learned latent generator. This provides a structured transition from a non-ODE kernel representation to an ODE-inspired kernel operator whose latent temporal evolution satisfies the composition principle of evolution semigroups. For time-dependent forcing terms, a direct kernel-based evaluation requires repeated quadrature over delayed kernel evaluations. To address this computational bottleneck, we derive a latent-first reformulation of the source term: the forcing field is projected onto the kernel-induced latent space, where its accumulated effect is represented as the solution of a forced linear latent dynamical system. For a fixed learned kernel, this reformulation is exact. The resulting system can then be integrated using a \(φ_1\)-based exponential integrator, yielding a source-evaluation path that is structurally aligned with the learned latent propagator. The resulting framework combines geometry- and physics-conditioned low-rank kernel factorization with alternative source-evaluation paths based on either explicit Duhamel quadrature or latent exponential integration. Dissipativity constraints on the latent generator and identity-preserving corrections near the initial time are optional architectural components. Numerical experiments on advection-iffusion-reaction (ADR) problems compare the non-ODE and semigroup-aware formulations, evaluate the quadrature and \(φ_1\) source paths, and assess predictive accuracy and computational cost.
math.DS
For a time-homogeneous Markov process, the fixed-lag solution operator \( (T_τf)(x)=\mathbb E[f(X_{t+τ})\mid X_t=x] \) of the Kolmogorov backward equation evolves observables linearly and deterministically, even when the dynamics are stochastic. We approximate this operator using a two-stage method where we first approximate this operator directly from data, without knowledge of the governing equation (e.g., Fourier Neural Operator), and then compress the Galerkin projection onto a finite-dimensional observable space that is fixed in advance or selected variationally. The error analysis traces the learning error of the estimator, the sampling error of the Galerkin inner products, and the invariance and reconstruction properties of the observable space to the computed spectra and forecasts. The sampling error enters only through a multiplicative factor. Near an isolated spectral cluster, the invariance defect of a subspace controls its distance from the target invariant subspace. The forecast error is controlled by the one-step error of the reduced model and the projection error of the observable; when the measure is invariant and the reduced model preserves constants and is stable on mean-zero observables, the bound is uniform in the horizon. For deterministic dynamics, a learning rate for Fourier neural operators (FNOs) gives one explicit bound for the learning error. Numerical experiments illustrate spectral approximation and forecasting, including long-rollout stability and a comparison with extended dynamic mode decomposition (EDMD) when fewer transition pairs than basis functions are observed.
physics.plasm-ph
We develop a two-stage surrogate model framework to predict the linear and nonlinear properties of ITG-TEM turbulence from GENE gyrokinetic simulations. The first stage uses a hierarchical ensemble of stability and mode classifiers, along with global and mode-specific regressors, to predict the linear growth rate and real frequency. The second stage uses these predictions as linear spectra in a Fourier neural operator (FNO)-based nonlinear surrogate model. The nonlinear model predicts saturated potential and heat flux spectra, together with zonal flow quantities. Two linear drive scalars, the maximum and integral of the linear growth rate over a low-wavenumber range, are also supplied directly to the networks predicting scalar quantities. For the architecture studied here, comparisons with models omitting either or both input paths show that linear information can improve spectral and scalar predictions under extrapolation toward stronger linear drive.
cs.LG
In variable-step operator learning, agreement at observed step sizes does not by itself determine predictions at other temporal resolutions. We propose an operator whose step-size dependence is prescribed by a first-order Duhamel approximation of semilinear dynamics. It combines exact linear propagation with a local nonlinear function of the field and its derivatives, allowing evaluation below the observation interval without supervision densely sampled across step sizes. Under suitable conditions and at fixed spatial resolution, we prove that agreement between models at two step sizes controls their prediction differences throughout a prescribed range. We also give conditions for local identifiability of both components from two step sizes, supporting interpretation of the learned structure. A latent extension applies the same construction to learned fields. We evaluate the base model on two synthetic problems and the extension on two real datasets against several baselines.
cs.AI
Sparse-view computed tomography (CT) reduces the number of acquired projections, but conventional sampling strategies are designed primarily for image reconstruction rather than for downstream physical inference. Here, we investigate optimal task-aware projection selection for directly predicting hemodynamics from sparse CT angiography (CTA) measurements. Using computational fluid dynamics simulations of intracranial aneurysms, we generate synthetic fan-beam CTA projections and train a Fourier neural operator (FNO) to predict two targets: the spatially resolved time-averaged wall shear stress (TAWSS) field and the spatially averaged TAWSS. The FNO was coupled with optimal input selection for simultaneous optimal sampling and training. Our goal was to investigate if agentic AI could discover optimal sampling strategies that outperform current methods. We compared fixed uniform and golden-angle sampling with three task-aware optimal selection strategies: Gumbel-Softmax, a Soft top-K method, and an agentic AI selector. The methods were compared at different view selection budgets. A detailed analysis of the AI agent was performed to understand its reasoning and performance. For spatially resolved TAWSS, Agentic and Soft top-K achieve the lowest mean errors (Agentic performs slightly better) and substantially outperform fixed sampling under extreme sparsity. For spatially averaged TAWSS, optimized selection becomes important at extremely sparse acquisitions. Analysis of the selected masks reveals multiple distinct subsets concentrated around informative regions. Moreover, masks optimized for hemodynamic prediction yield poor image reconstructions, showing that task-optimal acquisition differs fundamentally from reconstruction-optimal acquisition. This work suggests the possibility of designing sparse acquisitions directly around downstream physical quantities.
math.OC
We introduce the Safeguarded Neurally-Accelerated Preconditioning (SNAP) framework for hybridizing nonlinear optimization methods with pre-trained neural operators (NOs). SNAP targets parametric optimization problems arising from variational formulations of nonlinear partial differential equations (PDEs), where many parameter instances require repeated solves. Inspired by multilevel line-search methods, SNAP operates in cycles, combining classical optimization steps with a search direction generated by a pre-trained NO. Since NO-generated search directions carry no a priori descent guarantee, SNAP employs an a posteriori safeguard by accepting only corrections that satisfy prescribed descent and acceptance conditions. We prove global convergence of the SNAP algorithm and show that it preserves the convergence rate of the underlying optimization method independently of the employed NO architecture. Moreover, we also show that when the NO corrections are sufficiently accurate, SNAP converges faster, with the improvement directly related to the NO training error. To demonstrate SNAP's efficiency in practice, we run numerical experiments on a diverse set of nonlinear parametric optimization problems, including parametric minimal surface, Allen–Cahn, and nonlinear Helmholtz problems. The efficiency of SNAP is demonstrated by comparing two NO architectures, Fourier neural operator (FNO) and Transolver, and by using SNAP as a standalone solver and as a nonlinear preconditioner. Our numerical results show a substantial reduction in the number of required iterations, while demonstrating robustness and generalization across a wide range of problem parameters and mesh resolutions.
cs.LG
Full-order surrogates for high-dimensional nonlinear flows are often unnecessary when applications require only selected degrees of freedom. We propose reduced-order models that bypass full-state reconstruction by deploying graph neural networks (GNNs) directly on coarse meshes. Applied to parameterized 2D/3D confined cylinder flows and an unevenly heated cavity over Reynolds/Rayleigh-number ranges, the models generalize to unseen conditions and accurately predict sparse probe locations and integrated quantities. Coarse-mesh GNNs match or outperform finer-mesh surrogates while reducing training time and memory, even for 3D chaotic dynamics. A dynamics-informed adaptive sampling strategy guided by frequency-spectrum variations reduces maximum errors by an order of magnitude versus uniform sampling at negligible cost. Coarse-mesh training also makes noise-augmented multi-step unrolling practical, improving long-horizon autoregressive predictions. The approach outperforms parametric DMD, sparse MLPs, U-Nets, and Fourier neural operators at lower cost, and enables system identification and state estimation from sparse observations. These results show that coarse-mesh representations, adaptive sampling, and noise-augmented unrolled training form an efficient framework for reduced-order modeling of parameterized unsteady flows without solver-resolution reconstruction.
cs.CE
We study whether the equilibrium of a drop on a chemically patterned wall, and the capillary forces that the pattern exerts on it, can be learned from the free energy alone, without solution data. We introduce DropFENO (Droplet Free-Energy Neural Operator), in which Fourier (FNO), tensor-factorised Fourier (TFNO) and wavelet (WNO) neural operators are trained in two and three dimensions on the diffuse-interface free energy of the drop, with its volume and centroid constraints imposed as penalties. The predictions are tested with the virtual-work (Hellmann–Feynman) theorem: the force exerted by the pattern on a held drop is a wall integral of the equilibrium field which, for a drop straddling a wettability step, reduces identically to the contact-line law . DropFENO reproduces this law and the energy landscape of a drop crossing a step, and reproduces the pinning of the receding contact line less accurately. The residual error lies in the soft modes of the capillary energy (small displacements of the interface and the contact line). This explains why the energy is predicted accurately while contact-line positions are not, and why on striped walls the operators return Cassie-like states instead of the edge-pinned states whose anisotropic apparent angles the reference minimiser shares with lattice Boltzmann simulations. On held-out patterned walls the median energy of the predictions lies $\NdEmedRandFnoTwo$ of the free-drop energy above the best known minimum (FNO, two dimensions), at about one hundredth of the cost of a per-instance minimisation, and a refinement by a classical solver started from the prediction returns the reference state in almost all cases.