Neural Fields

Machine-learning surrogate models offer a promising alternative to high-fidelity Computational Fluid Dynamics (CFD) simulations for aerodynamic analysis and design. However, constructing accurate surrogates for realistic aircraft configurations remain challenging due to complex geometries, multiple flow regimes, and limited training data. This work presents the methodology that achieved first place in the ONERA CRM Wall Distribution Regression Challenge, which focuses on predicting pressure and skin-friction coefficient distributions over the NASA Common Research Model wing-body-pylon-nacelle configuration under different operating conditions. The proposed approach formulates the problem as a conditional neural field mapping spatial coordinates, surface normals, and operating conditions to aerodynamic wall quantities. Fourier feature encoding, a relative squared error objective aligned with the challenge metric, ensemble learning, and $k$-fold cross-validation are progressively introduced to improve prediction accuracy and exploit the limited training data. Beyond presenting the final methodology, the paper documents the successive model design choices that led to the winning solution through a comprehensive ablation study and discusses several alternative approaches that were investigated but ultimately discarded. On the hidden competition test set, the proposed methodology achieves an overall score of 8.81, outperforming the strongest organizer-provided baseline, which achieved a score of 8.64, while requiring approximately three orders of magnitude fewer trainable parameters. These results illustrate that carefully designed coordinate-based neural fields constitute an efficient and robust framework for aerodynamic surrogate modeling on complex geometries under limited-data conditions.

Density-based topology optimization is typically structured as a nested sequence of material updates, structural analyses, and sensitivity assessments. While neural density parameterization and dual-field physics-informed approaches provide data-free alternatives, most existing methods represent density and displacement as coordinate fields and make limited use of the discrete relationships inherent in the finite element mesh. The present study introduces HGTO, a unified graph-based formulation that extends complete neural topology optimization from coordinate space to finite-element graph space. Element densities are parameterized on the element graph derived from the mesh, and the structural state is determined on the corresponding node--element hypergraph. Finite element kinematics, numerical quadrature, constitutive response, and force assembly remain explicitly defined operations within the differentiable computation. The material field and equilibrium state are therefore coupled through a common finite-element incidence structure. Numerical studies show compliance comparable to conventional density-based optimization at substantially lower computational cost than a representative coordinate-based dual-field neural method. The same coupled formulation accommodates high-resolution and irregular meshes, three-dimensional structures, finite deformation, and elastoplastic response.

Transient glacier simulations with higher-order ice flow require the repeated solution of a nonlinear problem as the geometry evolves. In the online mode of the Instructed Glacier Model, the velocity field is represented by a neural network whose weights are warm-started from the previous time step and updated with a few optimizer iterations. We show that supplying the network with inexpensive input fields derived from low-order ice-flow balances improves this online solver. Unlike residual-based physics-informed neural networks, which incorporate physics through governing-equation penalties in the loss, our approach leaves the governing energy objective unchanged, adding physical structure through the network inputs. Across three real-world glacier configurations, the enriched solver is markedly more robust to solver settings. On the two alpine cases, it also improves the tuned accuracy--runtime trade-off, reducing surface-velocity errors by factors of two to four at fixed runtime and reaching few-percent relative errors with only $10^4$--$10^5$ trainable parameters, far fewer than comparable raw-input baselines. A 300-year Aletsch simulation then completes in under one minute, and the larger Valais domain in about two minutes, on a single GPU---a budget once reserved for much simpler shallow-ice models. Gains are smaller for the fast marine-terminating glacier, where nonlocal stress coupling favors larger or spectral networks. More broadly, the results suggest that enriching a neural solver's inputs with reduced-order physics can make repeated higher-order solves much cheaper, with no training data and no offline training.

Accurate head modeling requires a stable yet expressive geometric representation. Existing Gaussian-based head avatars commonly rely on parametric templates (e.g., FLAME) for Gaussian initialization and deformation, but these templates lack personalized priors and struggle to represent structures such as hair and clothing. To address this issue, we propose MGAvatar, a Gaussian-mesh hybrid representation that jointly models geometry and appearance through two Gaussian-mesh binding modes. Specifically, we introduce vertex-bound Gaussians and constrain their learnable parameters, enabling progressive mesh deformation to represent complex head geometry, while a pose-dependent offset module accounts for non-rigid deformations. Once geometry is stabilized, MGAvatar switches to face-bound Gaussians for appearance modeling. To improve appearance consistency across novel poses and viewpoints, we introduce a view-conditioned neural color field that alleviates artifacts caused by independently optimized Gaussian colors. In addition, we design a Gaussian offset network to predict Gaussian offset maps in the observation space, providing greater flexibility for face-bound Gaussians to capture dynamic facial textures. Extensive experiments on multi-view and monocular videos show that MGAvatar outperforms existing methods in rendering quality, producing high-fidelity head avatars with rich texture details.

Neural fields (NFs) map continuous coordinates to signals such as color or density, but fast high-quality reconstruction from sparse observations remains difficult. Classical Neural Tangent Kernel (NTK) regression gives closed-form fits, yet it is fundamentally linear and cannot accumulate reusable task priors. We develop three algorithms that address these gaps. NTK-KIP learns a distilled support set of coordinates (and optional labels) so that a finite NTK can inpaint large missing regions from little observed data, yielding a compact non-linear representation instead of a raw kernel solve. MetaQuill meta-learns a shared initialization for an INR so that new scenes can be adapted by updating only a small task-specific weight offset, which provides true feature learning and a reusable prior. Finally, MetaQuill-KIP fuses both ideas: it seeds the task with a KIP-style non-linear warm start, then refines only that small offset around the meta-learned initialization. MetaQuill-KIP achieves high-PSNR reconstructions and semantically plausible inpainting under very sparse observations, while requiring only lightweight per-instance adaptation, whereas diffusion-style baselines typically depend on large pretrained generative priors and costly per-image tuning. This shows that NTK-driven neural fields can be made both non-linear and meta-learnable, narrowing the gap between analytic kernels and practical few-shot reconstruction.

Neural operators are fast, differentiable surrogates for physical simulation, but their accuracy often degrades when domain geometry, size, or operating conditions differ from training. Supervised adaptation can recover accuracy, but even a small target set requires costly high-fidelity simulations. We therefore ask how pretraining and transfer can be designed together to reduce this deployment cost. LatentDDM first pretrains a neural operator to predict fields on small subdomains. For a new setting, it freezes this operator and trains only a lightweight module that composes the local predictions. We evaluate our method on two complementary problems: steady Darcy flow, where long-range pressure coupling must extend across increasingly large porous domains, and unsteady incompressible flow around a pitching airfoil, where rollout errors compound as target pitching frequencies exceed the training range. Compared with the capacity-matched models that process the full domain at once, LatentDDM's error is 36-56% lower on larger Darcy domains after adaptation with 16 target simulations. It also improves 20-step field rollouts in fast-pitching airfoil flow, both zero-shot and after few-shot calibration. These results identify the co-designed local pretraining and composition-level transfer as a promising design principle for physical foundation models.

Physically consistent motion planning remains a fundamental challenge in embodied AI, as generated trajectories must strictly conform to real-world execution dynamics. While latent world models offer a promising approach by predicting these dynamics, existing methods learn unconstrained future representations where absorbed physics remains implicit. Therefore, they fail to form reusable physical knowledge, which compromises reliability in unpredictable open-world navigation. To address this, we propose a novel Energy-Structured Latent World Model (ELWM). Our key idea is to structure the ELWM latent state to explicitly carry energy and momentum, ensuring strictly causal transitions via dissipation and control ports. Trained on multimodal RGB-D and inertial interaction histories, our model guarantees physically consistent predictions. We further implement this for motion planning by constructing Physics-Conditioned Neural Time Fields (PC-NTF), a key technical cornerstone that integrates ELWM into an arrival time field via the Eikonal equation to yield a physically-informed navigation policy. Across held-out scenes, our evaluation reveals significant improvements. Compared to generic latent models, PC-NTF reduces 0.8-s motion-prediction NRMSE from 0.36 to 0.29. Against Active Neural Time Fields, it improves navigation success from 81.3% to 89.7% and SPL from 0.64 to 0.73, while cutting the physical collision rate from 12.1% to 5.8% and the Eikonal residual from 0.083 to 0.031. Beyond these targeted gains, our results demonstrate that embedding explicit physical structures into latent spaces intrinsically bridges the gap between predictive world models and safe, dynamically feasible motion planning.

A graph layout is normally a table of $N$ free coordinates. We optimise a function with a fixed number of parameters instead. This gives a drawing a sample complexity and an extensible domain. Force-directed algorithms remain the standard tools for graph drawing. The most accurate among them minimise stress in the Kamada-Kawai formulation by directly optimising the node coordinates, at a full objective cost of $O(N^2)$ in time and space. Here, we propose Fling (Field Layout via Implicit Neural Geometry), a small neural network mapping the distances of each node to a set of landmarks, positioning it in the plane by training on the layout energy. The full spring system then becomes tractable without its distance matrix, as rest lengths follow from a landmark bound in constant time per pair while a second network learns the majorisation sums from exact anchor rows, at $O(|\mathcal{A}|N)$ per step for $|\mathcal{A}|\ll N$ anchors. Unlike neural drawers that read the graph by message passing, we represent the drawing as a function of node features. An unseen node costs one forward pass, where sparse and low-rank majorisation remain transductive. As the unknowns are weights rather than coordinates, the energy only requires a small fraction of the nodes, and a field fitted that way outperforms PivotMDS, landmark MDS, and a kernel ridge trained on the same energy and features, when the task is fitting the energy of a graph from a sample of its nodes. In addition, the same parameterisation enables a stochastic pivot stress variant, an aesthetics-optimised variant carrying a neighbour-embedding energy with node-edge clearance and crossing terms on the same field, and conditioning on the weight between two energies gives a whole layout family from one run.

High-fidelity immersed-boundary simulation resolves the coupled motion of a deforming swimmer and its surrounding flow, but the resulting cost limits repeated evaluations for engineering design, parameter studies, and control. We develop neural-operator surrogates for temporal prediction of the hydrodynamic fields generated by planar and volumetric eel swimmers. The surrogates are trained on regular-grid fields exported from adaptive fluid--structure simulations and are conditioned on swimmer geometry and Reynolds number. The planar model jointly predicts two velocity components, scalar vorticity, and pressure. On five held-out high-Reynolds-number trajectories, its full-domain global relative L^2 error is 3.51 %. The volumetric formulation uses three target-specific models with a common multichannel input: one model predicts three-dimensional velocity, one predicts vorticity, and one predicts pressure. Their full-domain global relative L^2 errors on five held-out within-range trajectories are 3.44 %, 5.58 %, and 19.2 %. Together, the results demonstrate the feasibility of field-resolved neural surrogates for moving-boundary swimmer flows while identifying pressure accuracy and physical consistency as priorities for further development.

The ability to accurately assess and anticipate risks in safety-critical scenarios is crucial for autonomous driving systems. While existing research has made progress in collision prediction, accurately quantifying risk levels from monocular vision inputs remains challenging due to the complex dynamics of multi-agent interactions and the inherent uncertainty in real-world environments. To address these challenges, we present NSF-HRPT, a novel framework that combines learning-based perception with structured reasoning for quantitative risk assessment. Our approach features a Neural Semantic Field (NSF) that learns to model scene semantics, trajectory predictions, and probabilistic Time-to-Collision (TTC) distributions from simulation data. During inference, the pre-trained NSF serves as a prior for our Hierarchical Risk Perception Tree (HRPT), which enables efficient parallel computation and spatial reasoning about multi-agent risks. Additionally, we introduce a Sim2Real enhancement strategy that improves real-world applicability without retraining by incorporating priors from foundation models. Extensive evaluations demonstrate that our framework achieves state-of-the-art performance on synthetic benchmarks and delivers competitive, near-state-of-the-art results on real-world datasets for both TTC estimation accuracy and risk localization precision. The proposed method provides an effective solution for real-time risk awareness from monocular camera inputs.

Designing an effective electromagnetic inverse-scattering solver requires faithful enforcement of nonlinear full-wave physics together with an expressive prior on the unknown permittivity contrast. We propose ScoreField, a neural inverse scattering framework that integrates coupled implicit neural representations (INRs) with a pretrained score-based generative prior. ScoreField employs two INRs to parameterize the permittivity contrast and the induced current fields, and jointly optimize them under the Lippmann-Schwinger equations. In addition to the implicit regularization by the INR architecture, the score model provides a learned prior gradient on the contrast, which is propagated to the contrast INR through the chain rule. This formulation enables ScoreField to effectively handle strong multiple scattering, where nonlinear wave interactions require accurate modeling of the coupled full-wave physics. We evaluate ScoreField on simulated weak- and strong-scattering benchmarks, the canonical Austria phantom, and experimental Fresnel measurements. We note that ScoreField significantly improves reconstruction fidelity and suppresses artifacts relative to classical full-wave methods and deep learning baselines, achieving an average PSNR improvement of $1.8 \, \mathrm{dB}$ over the best competing method on real Fresnel data.

We propose a neural representation for minimal surfaces. Unlike prior approaches based on discretization or Physics-Informed Neural Networks (PINNs), where meshes or neural fields are optimized to approximate the governing equations, our method builds on an exact representation, similar to the classical Weierstrass--Enneper parameterization, yielding minimal surfaces up to negligible quadrature error in evaluation. We formulate a training objective for the Plateau problem that optimizes over this representation.

Neural surrogates are widely used in scientific machine learning for fast prediction of three-dimensional (3D) thermo-fluid fields. However, generating training data using conventional numerical solvers often incurs substantial computational and storage costs. We propose to train an attention graph neural network by minimizing the finite-volume method (FVM) residuals of the governing equations. These residuals are evaluated directly on the mesh, requiring no labeled data. We evaluate the trained surrogates against computational fluid dynamics (CFD) references and a data-supervised baseline across four scenarios. On the two steady-state benchmarks, the FVM-loss model achieves an all-field normalized root-mean-square error (nRMSE) of 2.3-2.8%. It demonstrates close agreement with the CFD references, including the buoyancy-energy coupling. On the two parametric transient cases, the FVM-loss model outperforms the supervised baseline in terms of accuracy, while avoiding the data-generation cost entirely. These results indicate that the FVM loss can provide a practical training signal for neural surrogates and reduce the model development cost.

Inferring latent physical properties from sensory observations is a fundamental challenge in machine perception. Among available sensing modalities, thermal imaging is particularly promising because temperature evolution is directly governed by heat-transfer physics and therefore encodes information about underlying thermophysical properties of a scene. Recovering spatially resolved thermophysical properties from thermal observations could transform applications ranging from digital twins and infrastructure monitoring to robotics and scientific imaging. However, existing thermal scene reconstruction methods can recover temperature fields in complex 3D environments without identifying the thermophyiscal properties that govern thermal evolution, whereas inverse methods provide physically interpretable parameter estimation but typically rely on simplified geometries and controlled experimental conditions. Here we introduce ThermoField, a framework that unifies thermal scene reconstruction and thermophysical parameter estimation through differentiable heat-transfer simulation. The proposed framework represents these quantities as spatially varying neural fields and constrains them through scene geometry, governing heat-transfer physics, and temporal thermal observations. We demonstrate that ThermoField jointly reconstructs geometry, estimates spatially varying thermal diffusivity, and predicts thermal evolution under previously unseen environmental conditions. By integrating neural scene representations with differentiable heat-transfer solver, the framework enables physically interpretable parameter inference in complex 3D scenes. Our results establish a bridge between thermal scene reconstruction and inverse heat-transfer analysis, providing a unified approach for geometry reconstruction, thermophysical property estimation, and predictive thermal simulation from thermal observations.

A learning-based physics-constrained neural kernel for sound field estimation is proposed. Sound field estimation aims to estimate the spatial distribution of an acoustic field from a discrete set of microphone measurements, which have a wide range of applications. Among existing sound field estimation methods, kernel-regression-based methods offer a flexible and principled framework for incorporating physical constraints and allow inference through linear operation. It is also possible to adapt the kernel function to the target acoustic environment by representing the directional weighting function as an implicit neural representation (INR) and optimizing hyperparameters using measurements. However, the kernel function is generally optimized for single snapshot measurements of the microphones, which can lead to strong overfitting and poor generalization. We propose a source-position-dependent INR for the directional weighting function, enabling the kernel function to capture common directional patterns and to generalize to unseen source positions in the target acoustic environment. Experimental results indicate that our proposed method outperforms the snapshot-based method by estimating a directional weighting function that matches the directivity of the target sound field.

The Minkowski functionals of a field's excursion sets -- area, boundary measure, and Euler characteristic -- describe its level-set morphology; the Euler characteristic is the cheapest handle on topology. We derive smooth Monte-Carlo estimators for all three of a continuous neural field, evaluated at scattered points via the co-area formula and Gauss-Bonnet, using only autodiff: no grid, no complex, no persistence. The estimator is accurate to 1-3% against exact topology in 2D and 3D, and costs about 3 ms per iteration where a persistent-homology (PH) loss on a cubical grid costs 650-1000 ms -- a 250x gap. We establish four design rules without which these losses silently fail: a dense level ladder (invariants are flat in the parameters away from transitions), a $C^2$ backbone (ReLU nets hide curvature in kinks), the full Minkowski vector (Euler characteristic alone is an alternating sum, gamed by debris-hole cancellation; pricing perimeter closes the channel), and sampling-scale coverage. In 2D the vector-valued cap is the only method in a controlled comparison that both repairs topology (3/3 seeds) and preserves fidelity -- uniform smoothing repairs at 11-17x the fidelity cost, and the Euler term alone repairs nothing. In 3D neural-SDF fitting, however, a failure mode we believe general to any sampled soft topology objective appears: gradient descent adversarially hides topological noise below the sampling density, where the estimator is blind -- spurious-feature counts are invariant to 4x more samples, and closing the window needs cubically many points, erasing the cost advantage. A grid-based PH baseline, whose complex is the evaluation resolution, solves the same benchmark ($4/9$ exact; median $b_1$ error 1 vs. ours above $10^4$). The 250x cost of persistence is, at present, the price of having no null space. We release estimators, receipts, and benchmarks.

Quantifying plant growth dynamics from sparse longitudinal 3D observations is fundamental for agriculture and plant sciences. Yet, plants pose unique challenges: they undergo intricate non-rigid deformations, exhibit changing topology as new organs emerge, and often lack explicit temporal correspondences between consecutive data acquisitions due to newly formed tissue. Methods designed for general scenes struggle to model topology changes and asynchronous organ growth characteristic of plants. To address these challenges, we introduce GrowFields, a compositional dynamic neural field representation for organ-aware 4D plant growth modelling from point cloud time series. Our approach decomposes a plant into its constituent organs and aligns each organ into its own canonical coordinate frame, isolating intrinsic growth patterns from global plant motion. We then learn a shared continuous neural deformation field that models temporal dynamics across all organs, conditioned on learnable per-organ latent codes capturing organ identity and growth characteristics. The resulting modular yet unified representation naturally accommodates the asynchronous development of plant organs while remaining grounded in the practical setting of organ-level plant tracking. We evaluate GrowFields on growth sequences from four plant species, assessing geometric fitting and organ tracking accuracy using manually annotated leaf-tip trajectories. Results demonstrate consistent improvements in spatial precision, temporal coherence, and morphological fidelity over a range of existing representations.

Self-collision remains a persistent challenge in SMPL-based human pose estimation and motion generation. Under extreme articulations or stochastic motion synthesis, generated meshes frequently exhibit self-penetrations, leading to physically implausible results. We propose PoseShield, a neural collision constraint defined directly in SMPL pose space. We formulate collision correction as a constrained optimization problem and connect the learned constraint with the Eikonal equation. Enforcing Eikonal regularization ensures non-vanishing gradients near the collision boundary, improving numerical stability and robustness of the optimization process. Unlike prior methods that operate in the mesh space or rely on heuristic penalties, our approach operates directly in the low-dimensional space of human poses and is theoretically grounded. The same learned constraint extends to human motion sequences, providing a generator-agnostic post-hoc collision corrector without retraining the underlying motion model. Experiments on a newly constructed SMPL pose benchmark show that our method achieves a 95.8% success rate and outperforms state-of-the-art baselines.

Physics-informed neural networks (PINNs) have recently emerged as a promising framework for addressing the Calderón inverse problem from limited boundary data. In this work, we revisit neural Calderón inversion by introducing multiscale boundary excitations based on randomized wavelet functions and investigating the role of Fourier-feature encoding (FFE) for representing sharp conductivity variations. We propose a physics-informed reconstruction framework that represents the unknown conductivity and the associated family of electric potentials with separate neural networks conditioned on the applied boundary excitations. The governing elliptic PDE is enforced through physics-informed residuals, while finite Dirichlet-to-Neumann (DtN) data are incorporated through boundary losses. Using synthetic data from a finite-difference forward solver, we evaluate the method on conductivity fields with inclusions, sharp interfaces, smooth profiles, and heterogeneous media. Results show that the framework recovers dominant conductivity structures from finite boundary measurements with relative errors between $3\%-12\%$ approximately. We show that FFE improves the reconstruction of localized sharp features, particularly for inclusions and interfaces, but are not universally optimal, with raw-coordinate networks performing competitively for smoother fields. These results highlight coordinate representations and boundary excitation design as key factors in neural Calderón inversion.

Phase-field models play a central role in the continuum description of phase separation, in which the bulk free-energy density and the interfacial thickness parameter determine pattern formation and microstructural evolution. In practice, these constitutive quantities are rarely known a priori and must be inferred from limited dynamical observations. In this work, an extended pseudo-spectral physics-informed neural network (ESPINN) framework is developed for the inverse identification of phase-field models from transient snapshot data. It enables the simultaneous recovery of both the bulk chemical potential and unknown gradient coefficients. Numerical experiments on the one-dimensional Cahn-Hilliard equation demonstrate accurate and statistically stable reconstruction in the noiseless regime, with substantial constitutive information recoverable from even a single snapshot pair. In the presence of noise, reconstruction accuracy degrades gracefully, and increasing the number of snapshots improves robustness by reducing variance across runs. These results establish ESPINN as a data-efficient and physically consistent approach for learning free-energy structure in continuum models of phase separation.

Noise in neural computation is typically regarded as a disturbance, but its spatial distribution may also actively regulate which parts of a network participate in computation. This paper investigates the spatial partial functionalization of Noise-modulated Neural Networks using noise fields. We first present an activation function suitable for this goal, the crossing activation function, using the sample-level, statistical-level, and analytical-level implementations, and examine parameter reuse across these implementations. We then introduce a virtual noise field, an auxiliary continuous space for generating spatially structured network noise fields that activate partially overlapping subnetworks. Using one-dimensional function approximation tasks, we evaluate how multiple functions can be stored in a single network when each function is assigned to a different noise-field location. The results show that memory capacity improves when the spatial arrangement of noise fields reflects the proximity relationships among the functions to be learned, whereas mismatches in noise field structure can reduce effective capacity. These findings suggest that structured noise can serve not only as a perturbation but also as a topology-defining factor for functional subnetwork selection.

Mean field games efficiently approximate a very large population of strategic agents. While these games can aid the understanding of complex systems, their deployment in real-world settings is challenged by the specification of their parameters: mean field games (MFGs) often involve hidden preferences, constraints, and interactions that can rarely be theoretically derived or directly observed. To address this gap, we present a neural network-based framework for learning parametric, finite-state MFGs from observed population dynamics. To do so, we formulate the parameter calibration as an inverse problem and use implicit differentiation to backpropagate through the games' equilibrium. The resulting approach is fully differentiable and enables us to estimate flexible trajectory-wise parameter paths, including state- and time-dependent specifications without requiring observations of the individual agents' actions or rewards. We provide a proof for the exactness of the gradient computation in a discrete-time formulation. We validate our framework through numerical experiments across four systems of increasing complexity, ranging from synthetic linear-quadratic benchmarks to real-world urban mobility datasets.

Photonic quantum machine learning offers a route to trainable physical representations built from phase, interference and measurement. However, its role in scientific machine learning remains largely unexplored. Physics-informed neural fields provide a natural setting, because differential equations require trial spaces that preserve phase, frequency and derivative structure. Here we introduce a photonic quantum neural field in which coordinates become trainable optical phases, are mixed by multi-photon Fock-space interference and are decoded from photon-number measurements. The photonic circuit is optimized as the neural-field representation itself, not as a fixed feature map or hardware accelerator. Photonic measurement is therefore a trainable representation on which the physics-informed residual is minimized. Across seven elliptic, wave, nonlinear dispersive and inverse PDE benchmarks, we observe a phase-complexity transition: classical coordinate and Fourier-feature networks suffice in smooth regimes, whereas the photonic field is most accurate when residual derivatives amplify phase mismatch. In the hardest regimes it gives the lowest errors, with margins reaching an order of magnitude and about one quarter of the trainable parameters of classical baselines. Frozen and shuffled controls, together with noise stress tests, attribute this gain to learned interference and stable Fock-probability readout under compound perturbations. These results identify photonic quantum measurement as a representation-learning principle for scientific machine learning.

Physics-grounded video generation requires controllable 3D object dynamics that remain physically consistent under contact, deformation, and external forcing. Existing trajectory-based methods often model isolated physical effects, making it difficult to compose conservative and non-conservative dynamics in contact-rich 3D scenes. We present NEXUS, a neural energy-field framework for contact-rich 3D object dynamics. NEXUS represents each object as a structural graph and constructs dynamic object-object and object-environment contact graphs. Inspired by Hamiltonian Neural Networks, NEXUS formulates motion through scalar energy and dissipation terms rather than directly predicting states or accelerations. Conservative effects, including gravity and elastic deformation, are composed as additive energy terms, while non-conservative effects such as damping and impact-induced energy loss are modeled with learned Rayleigh-style dissipation. Forces are derived by differentiating the energy and dissipation functions and rolled out with a multi-substep semi-implicit integrator. Across controlled trajectory benchmarks, NEXUS improves long-horizon accuracy over representative learned and physics-structured dynamics baselines under varying mechanical properties and physical-effect compositions. We further show that NEXUS trajectories provide effective guidance for contact-rich video generation, improving physical plausibility while maintaining competitive visual quality.

Reconstructing local stress fields in heterogeneous microstructures under non-linear, history-dependent loading remains a major computational bottleneck in multi-scale simulations. We propose a coupled LSTM-GNN framework that links the temporal and spatial aspects of local stress field reconstruction. A Long Short-Term Memory network encodes macroscopic stress-strain sequences into a compact hidden state that captures the path-dependent constitutive response, while a physics-informed Graph Neural Network reconstructs the spatially-resolved stress field at each time step. We introduce a relative weighting strategy with linear warm-up to balance the data-driven reconstruction loss and a discrete divergence-based equilibrium penalty. This resolves the scale mismatch that prevents fixed-weight formulations from converging in the elasto-plastic regime. The model is trained on 10,000 non-proportional loading paths applied to a periodic plate-with-a-hole microstructure and von Mises elasto-plasticity. The model achieves three orders of magnitude speedup over finite element simulations and generalizes to loading sequences twice the training length, with 1.9% cumulative error. Because the graph relies on mesh connectivity instead of the specific element type, one trained surrogate can be applied directly without retraining to meshes with different element types and to both coarser and finer resolutions, while in all cases reproducing the high-fidelity quad-element FE field used during training. Indeed, the message passing characteristics inherent to GNN and MeshGraphNet architecture render the model mesh-agnostic. Analysis of the LSTM hidden states suggests a low-dimensional structure related to the internal state variables of the constitutive model.

Neural fields parameterize data as functions from coordinates to values, providing a unified framework for representation learning across modalities. Existing approaches are dominated by per-sample meta-learning, which scales poorly due to memory-intensive inner-loop optimization. The natural alternative -- feed-forward encoding -- typically introduces modality-specific assumptions, sacrificing the generality that makes learning with neural fields attractive. We argue that locality and hierarchy are useful priors for learning field representations that can be injected without compromising modality-agnosticism. We propose LH-NeF, a framework to learn general-purpose tokenized representations of continuous signals. A locality-preserving hierarchical encoder maps raw coordinate-value field observations to structured tokens, from which the field is reconstructed during training. By replacing meta-learning's inner loop with a single forward pass, LH-NeF uses 42$\times$ less memory and supports 133$\times$ larger batches than the strongest modality-agnostic baseline. Across images, 3D shapes, and climate fields, our learned representations match or exceed performance of modality-agnostic, modality-specific, and specialized generative neural field baselines on both reconstruction and downstream tasks.

Deformable 3D Gaussian Splatting (D-3DGS) re-constructs dynamic scenes from monocular video by deforming a canonical set of 3D Gaussians through a positional-encoded MLP of frame time t. Although fitted to a continuous variable, the MLP couples no two values of t in its architecture and effectively predicts discrete per-frame offsets, leaving temporal smoothness to emerge only as a byproduct of optimisation. We redesign the deformation field as a stack of Closed-form Continuous-time (CfC) cells, a Liquid Neural Network (LNN), that is the closed-form solution of the Liquid Time-constant ODE while preserving every other part of the D-3DGS pipeline. Each cell exposes a sigmoidal time gate that interpolates between two candidate hidden states, baking a learned smooth response to t into the loss landscape without invoking any numerical solver. On the eight D-NeRF and seven NeRF-DS scenes the liquid field matches or exceeds the MLP baseline in aggregate, with its largest gains concentrated on the scenes with the most high-frequency articulated motion. The result is a near-zero-friction architectural design that turns the discrete MLP deformation field into an explicit continuous-time function of t.

Real-time monitoring of the temperature distribution within components and sub-structures is a challenging topic in many systems due to restrictions on feasible sensor locations. While machine learning (ML) proves a versatile tool in many applications, its adoption for high-resolution thermal monitoring is hindered by the availability of high-quality datasets for training. In this work, we propose a novel approach for generating datasets for industrial applications based on randomized physics-based simulations. We demonstrate the approach in a proof-of-concept hardware setup: A neural network (NN) trained only on such a synthetic dataset, is used to reconstruct the internal temperature field from sparse sensors embedded in the hardware. The NN-based reconstructions do not only outperform Kriging in robustness but also enable real-time inference, making the method suitable for online monitoring of otherwise unobservable thermal states.

In this paper, we propose a reconstruction framework that leverages the Wavelet Scattering Transform (WST) as a multi-scale feature extractor to impose statistical priors under sparse observation conditions. The reconstruction problem is formulated as an optimization task and solved using a neural field, with the WST incorporated into the training loss function. As a proof of concept, we validate the proposed method on HRTF upsampling. A masking strategy is applied to the WST coefficients, resulting in a two-phase procedure. The first phase learns a binary mask from a small multi-subject dataset, while the second phase applies the learned mask to the WST coefficients of an individual HRTF to preserve informative statistical structures during reconstruction. Validation against baseline methods, which also serve as an ablation study of the different components of the framework, demonstrates the effectiveness of the proposed approach.

3D representations are fundamental to scene rendering, understanding, and interaction. Recent approaches, such as 3D Gaussian Splatting and Neural Radiance Fields, achieve impressive photorealistic novel-view synthesis, but lack the ability to easily decompose scene elements into a few primitives, requiring additional segmentation or grouping for object-level manipulation. We present MLP-Splatting, a method that enables scene decomposition via a few expressive light-field primitives while providing photorealistic novel-view synthesis. MLP-Splatting models each primitive as an independent compact MLP with localized spatial support that predicts radiance and opacity. In contrast to low-level Gaussian primitives or a single global radiance field, our neural primitives provide greater expressive capacity while remaining spatially localized. Rendering is performed through efficient sparse volumetric compositing over ray-primitive interactions. Our primitives are supervised using RGB supervision alone, which yields primitives that represent local scene regions often corresponding to objects or object parts, enabling interactive object-level editing without segmentation masks by selecting a handful of primitives. Our method, augmented with optional semantic feature distillation, enables open-vocabulary scene interaction and open-set instant segmentation. Compared to state-of-the-art methods, we achieve substantially lower memory usage (1/15$\times$) and faster rendering (3$\times$), as we show in our experiments compared to semantic 3DGS methods. Project Page: https://shinjeongkim.com/mlp-splatting

Neural field surrogates can accelerate photonic design loops, but a surrogate that looks accurate in global field error can still mis-rank candidate devices when the final decision depends on localized output-port readouts. This risk is acute in propagation-dominated MMI splitters and couplers, where port power, splitting, phase, and coupling are determined by accumulated modal interference and output-window aggregation rather than by average field similarity alone. We study this field-to-design mismatch through a Field/Mediator/Readout view that separates dense complex-field error from propagation-profile and output-window errors before port aggregation. To align the surrogate with this chain, we propose PaNO, a propagation-aligned neural operator that keeps the full-field prediction interface while organizing latent states around local boundary structure, transverse modal content, axial propagation, and cross-mode interaction. We also evaluate PaNO-R2, an output-aware feedback variant for residual field components near the port region. On a 15-wavelength tunable $3{\times}3$ MMI benchmark with 4608 held-out fields, PaNO lowers NeurOLight's port-power error from 0.2018 to 0.0739 despite slightly higher cMAE, showing that global field accuracy alone is not sufficient for design-relevant readout fidelity. PaNO-R2 attains the best cMAE, propagation-profile error, output-profile error, and port-power error, reducing NeurOLight's port-power and output-profile errors by 72.7% and 72.5%.

3D Gaussian Splatting (3DGS) renders new views of a scene in real time. Like every rasterizer, it answers only primary rays, the rays from the camera through the image. It cannot trace the secondary rays that shadows, ambient occlusion, and global illumination need. We turn a trained 3DGS scene into a ray oracle by distilling a Directed Distance Function (DDF). The DDF is a small neural field. It takes a ray, given by an origin and a direction, and returns the distance to the first surface and whether the ray hits anything. Each query is one forward pass. The field is 52MB, and its size does not depend on the number of Gaussians, so its cost and memory stay flat as the scene grows. We make three points. First, we study what supervision a DDF needs. Depth rendered from the Gaussians is too blurry to teach thin parts, while clean distance supervision recovers them. Second, we measure speed. The DDF is 26 to 72 times faster than sphere tracing an equivalent signed distance field, and unlike a bounding volume hierarchy built over the Gaussians, even on dedicated RT-core hardware, its query time and memory do not grow with the scene. Third, we show a pipeline that needs no mesh: images give a 3DGS scene, a neural surface gives clean distances, and the DDF learns from them. We use the DDF as a secondary-ray oracle for global illumination. It reproduces reference ray-traced shadows at 30.3dB and ambient occlusion at 21.3~dB across 142 objects, and on real captured scenes. Our codes are available at https://github.com/smlab-niser/ddf-gs.

Efficient onboard multi-field sparse reconstruction is essential for the autonomous operation of aerospace vehicles. While existing deep learning models exhibit promise for single-field reconstruction, deploying multiple independent models leads to prohibitive model size growth and fails to exploit cross-field correlations, particularly under few-shot conditions. To address these challenges, we first propose a lightweight multi-task Fourier neural operator (MTL-FNO), an end-to-end joint training framework based on hard parameter sharing. In each layer, the parameters are divided into shared and task-specific components to capture common features across fields while preserving task-specific characteristics. Moreover, the task-specific fine-tuning parameters are implemented as low-rank terms, achieving substantial model compression. Second, to address the difficulty of co-optimizing shared and task-specific parameters along with their real and imaginary parts, we revisit the FNO's spectral weight from a polar-form perspective and devise a physically meaningful decoupled optimization scheme. Specifically, we apply polar decomposition to slice-wise disentangle the spectral weight into a unitary tensor encoding phase information and a positive semi-definite tensor characterizing amplitude. By decoupling the optimization of phase and amplitude, our method can effectively mitigate tasks conflict. Meanwhile, to preserve unitary geometric fidelity during training, the Cayley transform is introduced to reparameterize the unitary tensor, converting the constrained optimization problem to an unconstrained one. Finally, the effectiveness of the proposed method under few-shot conditions is validated on two representative engineering cases. Results show that MTL-FNO achieves accuracy comparable to or even surpassing that of standard FNO, while reducing total model size by 76% and 60%, respectively.

Radars are an ideal complement to cameras: both are inexpensive, solid-state sensors, with cameras offering fine angular resolution, while radars provide metric depth and robustness under adverse weather. However, radar data is more difficult to interpret than camera images and varies significantly between sensors, necessitating increased reliance on simulation for prototyping sensors and processing pipelines. Recent work treating radar reconstruction as a novel view synthesis problem has shown great promise in reconstructing radar-relevant geometry and simulating low-level radar data. However, such methods are constrained by the low spatial resolution of the underlying radar. To address this, we propose a unified differentiable renderer, RadarSim, which leverages the high angular resolution of RGB cameras to generate Doppler radar range images from a camera-initialized neural field. Using a novel data set of calibrated radar camera recordings from a custom hand-held rig, we demonstrate that RadarSim produces sharper geometry and Doppler range frames than radar-only reconstructions.

We introduce "Courant", a Perceiver-based encoder-processor-decoder surrogate model that has latent features exhibiting adaptive specialization and local support in the physical space, enabling functionality akin to an adaptive hp-refinement scheme, an attribute that is highly desirable in traditional numerical solvers and scientific machine learning broadly. The proposed architecture combines a shared random Fourier feature coordinate embedding, state-adapted latent queries, and a light-weight decoder. Courant is trained end-to-end with steady or transient simulation data and only a standard L_2 prediction loss in the physical space, achieving competitive accuracy on benchmarks. We demonstrate that Courant's inductive biases yield latents that are interpretable by design: they develop multiscale geometric specialization in the simulation domain and track coherent structures in the time-dependent case, acting analogously to time-evolving spatial basis functions and allowing for decoding a compact, geometry-anchored, partition-of-unity-like decomposition of the simulated field.

Reconstructing people, objects, and their interactions in 3D is a long-standing goal for intelligent systems. Often the input is RGB video from a moving camera, making the task ill-posed; depth is ambiguous, humans and objects occlude each other, and camera and object motion entangle to create apparent motion. Most prior work addresses humans or objects in isolation, ignoring their interplay, or assumes known 3D shapes or cameras, which is impractical for real-world applications. We develop RHINO (Reconstructing Human Interactions with Novel Objects), a three-step framework that recovers in 3D a human, novel (unseen) manipulated object, and static scene in a common world frame from a monocular RGB video. First, we leverage 3D-aware foundation models to obtain cues that stabilize Structure-from-Motion (SfM) even for low-texture regions; this yields a coarse shape and apparent motion of a manipulated object from foreground pixels, and a coarse scene shape and camera motion from background pixels. Second, we estimate a human in the camera frame via an off-the-shelf method, and subtract the camera motion from apparent motion to extract the object motion; this registers the human, object, and coarse scene shapes into a common world frame. Third, we refine shapes using a compositional neural field with per-component signed-distance fields. The latter further enables differentiable contact priors that attract surfaces while penalizing interpenetration, improving the physical plausibility of the final reconstruction. For evaluation, we capture a new dataset of handheld monocular videos synchronized with a volumetric 4D capture stage, providing ground-truth shape and camera motion. RHINO outperforms state-of-the-art baselines on novel-view synthesis and 4D reconstruction. Ablations show that each stage contributes substantially. Code and data are available at https://lxxue.github.io/RHINO.

Natural images are continuous, yet most generative models synthesize them on discrete grids, limiting resolution-flexible generation. Continuous neural fields enable resolution-free rendering, but prior methods introduce continuity only at the decoding stage as an interpolation module, leaving the generative latent space discretized and reconstruction-oriented. We propose RaPD (Resolution-agnostic Pixel Diffusion), which performs diffusion in a continuous Neural Image Field (NIF) latent space. RaPD bridges this reconstruction-generation gap with Semantic Representation Guidance for generation-aware latent learning and a Coordinate-Queried Attention Renderer for coordinate-conditioned, scale-aware rendering. A single denoised latent can be rendered at arbitrary resolutions by changing only the query coordinates, keeping diffusion cost fixed. Experiments demonstrate superior generation quality and resolution scalability.

Inverse problems in scientific sensing are often solved with either hand-designed regularizers or supervised networks trained on simulated labels, yet both can fail when the forward model is nonlinear, spectrally coupled, and physically delicate. We study this issue for noise sensing based on nitrogen-vacancy (NV) centers in diamond, where a quantum sensor measures magnetic-noise spectra generated by sparse spin sources. We show that replacing a common scalar/coherent forward approximation with a tensor power-summed dipolar operator changes the inverse landscape and exposes a center-collapse failure mode in free-density optimization. We propose NeTMY, an amortization-free coordinate neural field coupled to the differentiable NV forward model, with annealed positional encoding, multiscale optimization, sparsity/gating, and spectrum-fidelity losses. Across sparse synthetic reconstructions generated by the corrected operator, NeTMY achieves the best localization and distributional metrics in the tested benchmark. Mechanism experiments show that NeTMY does not directly execute the raw density-space gradient; its parameterization smooths and redistributes updates, mitigating the center-collapse pathology. These results position NV quantum sensing as a useful testbed for physics-faithful neural inverse problems.

Neural operators have become an effective framework for learning mappings between function spaces, yet most existing architectures realize operators within a single representational domain, such as physical, spectral, or latent space. In this work, we introduce UFO (Domain-Unification-Free Operator), a cross-domain neural operator framework that realizes operators through adaptive, jointly conditioned interactions among representations defined on distinct domains. UFO enables discretization decoupling: the input function can be observed at resolutions or locations different from those used during training, while the solution can be queried at arbitrary output resolutions. Across four complementary benchmarks covering discontinuous inputs, irregular sampling with spectral mismatch, nonlinear dynamics, and stochastic high-frequency fields, UFO delivers accurate, robust, and physically coherent predictions under distribution shifts. These results establish cross-domain, phase-modulated realization as a powerful framework for discretization-decoupled neural operator learning.

Full-vehicle crash simulations are computationally expensive, limiting their use in iterative design exploration. This work investigates learned hybrid surrogate models (MeshTransolver, MeshGeoTransolver, and MeshGeoFLARE) for predicting time-resolved structural deformation fields in an industrial lateral pole-impact benchmark. We evaluate whether neural surrogates can reproduce full-field crash kinematics with sufficient accuracy, spatial regularity, and structural plausibility for engineering interpretation. The proposed architectures combine local mesh message passing, geometry-aware global attention, and sparse contact-aware correction for autoregressive crash rollout. We compare mesh-based graph neural networks, attention-based geometric models, and hybrid architectures under a common training and hyperparameter configuration. The hybrid models capture both short-range structural interactions and long-range deformation patterns, while a sparse contact-aware variant assesses the effect of dynamic proximity interactions during rollout. On a 25-sample full-vehicle test set, the best hybrid model achieves a temporal mean root-mean-square error of 3.20 mm. While geometry-aware attention baselines are quantitatively competitive, qualitative side-view inspection shows they can introduce local spatial noise and deformation irregularities that complicate structural interpretation. In contrast, hybrid mesh-attention models provide the best balance between scalar accuracy, survival-space consistency, and physically interpretable displacement fields. These results suggest that crash surrogate assessment should combine global error metrics with downstream safety-relevant quantities and qualitative field inspection. The proposed methodology enables fast full-field predictions while preserving essential structural information for industrial crash-engineering analysis.

The remarkable achievements of both generative models of 2D images and neural field representations for 3D scenes present a compelling opportunity to integrate the strengths of both approaches. In this work, we propose a methodology that combines a NeRF-based representation of 3D scenes with probabilistic modeling and reasoning using diffusion models. We view 3D reconstruction as a perception problem with inherent uncertainty that can thereby benefit from probabilistic inference methods. The core idea is to represent the 3D scene as a stochastic latent variable for which we can learn a prior and use it to perform posterior inference given a set of observations. We formulate posterior sampling using the score-based inference method of diffusion models in conjunction with a likelihood term computed from a reconstruction model that includes volumetric rendering. We train the model using a two-stage process: first we train the reconstruction model while auto-decoding the latent representations for a dataset of 3D scenes, and then we train the prior over the latents using a diffusion model. By using the model to generate samples from the posterior we demonstrate that various 3D reconstruction tasks can be performed, differing by the type of observation used as inputs. We showcase reconstruction from single-view, multi-view, noisy images, sparse pixels, and sparse depth data. These observations vary in the amount of information they provide for the scene and we show that our method can model the varying levels of inherent uncertainty associated with each task. Our experiments illustrate that this approach yields a comprehensive method capable of accurately predicting 3D structure from diverse types of observations.

Travel-time tomography forces a trade-off between mesh resolution and stability in which the regularizer choice dominates what can be recovered. We introduce MIMIR, a differentiable framework that represents the 2D velocity field as a Fourier-feature neural network, replacing the grid-based slowness vector with a continuous, infinitely differentiable function. Prior neural-field tomography has staircased smooth fields under total-variation (TV) priors or oscillated near interfaces under $L^2$ Laplacian smoothing. We adopt second-order total generalized variation (TGV$^2$) and parametrize its auxiliary vector field as a second neural network jointly optimized with the velocity field, eliminating the inner Chambolle-Pock primal-dual loop that classically dominates TGV computation. On three synthetic benchmarks (Gaussian, horizontally layered, curved-fault inspired by OpenFWI) using cross-well acquisition, 5% travel-time noise, and five seeds, MIMIR-TGV$^2$ ties a classical FMM-LSMR baseline with auto-tuned hyperparameters on the Gaussian ($p=0.134$, paired $t$-test) and significantly outperforms it on layered ($p<0.0001$, 44% RMSE reduction) and curved-fault ($p=0.0002$, 33% reduction). Replacing TGV$^2$ with TV degrades performance on Gaussian ($p=0.004$) and layered ($p=0.003$); curriculum-annealed TV improves Gaussian RMSE by only 5.4%, confirming that TV's staircase bias is intrinsic to the regularizer rather than a scheduling artifact. The results empirically validate the Bredies-Kunisch-Pock prediction that piecewise-affine priors are better suited to subsurface velocity recovery than piecewise-constant TV priors. We argue that the central design choice in physics-informed neural-field inversion is not the network architecture but the regularizer. The full pipeline reproduces in under one hour on consumer hardware.

Neural operators provide fast surrogate models for time-dependent partial differential equations, but their standard autoregressive use usually assumes that the instantaneous field $u(t,\cdot)$ is a complete state. This assumption fails for delay equations, distributed-memory systems, and other non-Markovian dynamics: two trajectories may agree at time $t$ and nevertheless have different futures because their histories differ. We introduce the History-Space Fourier Neural Operator (HS-FNO), a neural operator for delay and memory-driven PDEs formulated on the lifted state $u_t(θ,x)=u(t+θ,x)$, $θ\in[-τ,0]$. The key computational step is to decompose one history-state update into a learned predictor for the newly exposed future slice and an exact shift-append transport for the portion of the history window already known from the previous state. This avoids learning deterministic history coordinates, reduces the learned output dimension, and enforces the natural discrete history update. We test HS-FNO on five benchmark families covering delayed reaction--diffusion, spatial epidemiology, nonlocal neural-field dynamics, delayed waves, and distributed-memory closures. Across ten random seeds, HS-FNO attains the lowest aggregate one-step, history-space, and rollout errors among the principal baselines. The largest gain occurs in autoregressive prediction, where aggregate rollout error decreases from $0.241$, $0.188$, and $0.185$ for current-state, lag-stack, and unconstrained history-to-history operators, respectively, to $0.094$. The same model uses fewer parameters than unconstrained history prediction. These results indicate that enforcing the discrete shift structure of history-state evolution is an effective inductive bias for non-Markovian PDE surrogate modeling.

Neural surrogate models for computational fluid dynamics (CFD) are typically trained as forward operators that map explicit problem specifications, such as geometry and boundary conditions, to solution fields. This ties the model to the conditioning variables seen during training and limits reuse under boundary-condition shifts or local geometry changes. We propose to reformulate steady CFD inference as an inpainting problem: instead of training on explicit boundary conditions, we learn a self-supervised prior over velocity fields and impose boundary constraints only during inference by fixing known regions such as inlet, outlet or unchanged regions from previous simulations. To scale this idea to large 3D meshes, we introduce a local neighbourhood tokeniser that represents high-resolution velocity fields as compact spatial latent tokens and train latent flow-matching and masked-autoencoder models on these tokens. On intracranial aneurysm hemodynamics, our method reconstructs full velocity fields from sparse boundary context, outperforms supervised neural surrogates under boundary-condition and dataset shift and enables local geometry editing by reusing unchanged simulation context. These results suggest that viewing CFD inference as context-conditioned inpainting can turn neural surrogates from task-specific predictors into reusable flow priors.

Modern deep learning treats neural networks primarily as endpoint functions from inputs to outputs. Inspired by the shift from force to geometry in physics, we ask whether a network should instead be understood through the geometry of its internal propagation. We define a neural propagation field as the collection of hidden-state trajectories and local Jacobian operators across depth. Endpoint losses constrain only the boundary behavior of this field, leaving its interior geometry underdetermined. We show that endpoint-equivalent models can differ by orders of magnitude in trajectory and Jacobian structure, and introduce observable field metrics such as path sensitivity, solver consistency, and trajectory/Jacobian retention. In controlled teacher-flow and PDE systems, endpoint fitting fails to recover the underlying propagation law. In real multi-path tasks, field-aware objectives improve unseen-path generalization, OOD robustness, and calibration when aligned with the observation structure, but can collapse when over-constrained. In continual learning, field-preservation regularization complements replay and distillation: on Split CIFAR-100, DER++ with field preservation improves average accuracy, backward transfer, and field-retention metrics. These results identify propagation-field quality as a measurable and trainable property of neural networks beyond endpoint performance.

Conventional MRI reconstruction methods treat images and coil sensitivities as discrete objects, leading to high memory demands and limited structural awareness that hamper effective regularization. These limitations hinder accurate reconstruction in highly undersampled scenarios, such as dynamic 3D cardiac magnetic resonance (CMR). We introduce a discretization-free, memory-efficient, model-based framework for dynamic 2D and 3D MRI reconstruction from highly undersampled data. We represent magnetization and coil sensitivities as continuous objects -- differentiable functions -- using tensor products of univariate neural fields. This tensor product structure enables scalable optimization in high-dimensional spatiotemporal settings. Our method outperforms state-of-the-art model-based reconstructions in dynamic 2D and 3D MR settings, preserving structure and motion even under aggressive undersampling (e.g., acceleration factor 16).

In this work, a data-driven framework based on Phase-Field simulations data is proposed to highlight the capabilities of neural networks to ensure accurate low dimensionality reduction of simulated microstructural images and to provide time-series analysis. The dataset was indeed constructed from high-fidelity Phase-Field simulations. Analyses demonstrated that the association of auto-encoder neural networks and principal component analyses leads to ensure efficient and significant dimensionality reduction: 1/196 of reduction ratio with more than 80% of accuracy. These findings give insight to apply analyses on data from the latent dimension. Application of Long Short Term Memory (LSTM) neural networks showed the possibility of making next frame predictions; that makes possible the acceleration of Phase-Field simulation without the need of high computing resources. We discussed the application of such a framework on various areas of research. Different methods are proposed from the conducted analyses, in order to ensure dimensionality reduction, including auto-encoders, principal component analysis and Artificial Neural Networks, and time-series analysis, including LSTM and Gated Recurrent Unit (GRU).

Grid-based neural networks such as convolutional autoencoders are widely used in dimension reduction-based surrogate models for computational fluid dynamics. In recent years, the use of coordinate-based approaches like conditional neural fields has emerged. Their independence of the spatial discretization is a beneficial feature for various applications in computational fluid dynamics. This paper discusses the spatio-temporal prediction of aircraft ditching loads using a conditional neural field approach. The model is evaluated using two datasets for the dynamic loads of the fuselage of a DLR-D150 aircraft, one of which relates to a single fixed spatial discretization and the other that includes data from different discretizations. When paired with a long short-term memory (LSTM) network in the latent space, the neural field-based model achieves a spatio-temporal prediction accuracy for the first data set that is close to that of grid-dependent convolutional autoencoder-based models, and with significantly less parameters. Results for the second data set demonstrate the ability of the neural field-based approach to reconstruct ditching loads accurately for heterogeneous spatial discretizations. This allows for flexible use of training datasets generated for different geometries and/or discretizations, as well as the use of the surrogate model to predict loads for different configurations.

We present Mesh Field Theory (MeshFT) and its neural realization, MeshFT-Net: a structure-preserving framework for mesh-based continuum physics that cleanly separates the physics' topological structure from its metric structure. Imposing minimal physical principles (locality, permutation equivariance, orientation covariance, and energy balance/dissipation inequality), we prove a reduction theorem for mesh-based physics. Under these conditions, the physical dynamics admit a local factorization into a port-Hamiltonian form: the conservative interconnection is fixed uniquely by mesh topology, whereas metric effects enter only through constitutive relations and dissipation. This reduction clarifies what must be fixed and what should be learned, directly informing MeshFT-Net's design. Across evaluations on analytic and realistic datasets, physics-consistency tests, and out-of-distribution validation, MeshFT-Net achieves near-zero energy drift and strong physical fidelity (correct dispersion and momentum conservation) along with robust extrapolation and high data efficiency. By eliminating non-physical degrees of freedom and learning only metric-dependent structure, MeshFT provides a principled inductive bias for stable, faithful, and data-efficient learning-based physical simulation.

As the demand to integrate Artificial Intelligence into high-stakes environments continues to grow, explaining the reasoning behind neural-network predictions has shifted from a theoretical curiosity to a strict operational requirement. Our work is motivated by the explanations of autoregressive neural predictions on dynamic physical fields, as in weather forecasting. Gradient-based feature attribution methods are widely used to explain the predictions on such data, in particular due to their scalability to high-dimensional inputs. It is also interesting to remark that gradient-based techniques such as SmoothGrad are now standard on images to robustify the explanations using pointwise averages of the attribution maps obtained from several noised inputs. Our goal is to efficiently adapt this aggregation strategy to dynamic physical fields. To do so, our first contribution is to identify a fundamental failure mode when averaging perturbed attribution maps on dynamic physical fields: stochastic input perturbations do not induce stationary amplitude noise in attribution maps, but instead cause a geometric displacement of the attributions. Consequently, pointwise averaging blurs these spatially misaligned features. To tackle this issue, we introduce WassersteinGrad, which extracts a geometric consensus of perturbed attribution maps by computing their entropic Wasserstein barycenter. The results, obtained on regional weather data and a meteorologist-validated neural model, demonstrate promising explainability properties of WassersteinGrad over gradient-based baselines across both single-step and autoregressive forecasting settings.

We extend our gauge-covariant stochastic neural-field framework by promoting architecture-level parameters to slow stochastic variables evolving in function space. Our effective theory is formulated in terms of classical commuting fields and provides symmetry-constrained diagnostics of marginality and finite-width effects through the maximal Lyapunov exponent, the amplification factor, and dressed spectral kernels. On top of this dynamics, we introduce a Markovian evolutionary scheme compatible with the local $U(1)$ structure of the effective model. By using a minimal implementation, the genotype is reduced to the weight-variance parameter $σ_w^2$, and the fitness functional combines spectral agreement, marginal stability, and a symmetry-constrained critical anchor. Comparing three evolutionary models, we find that only the fully symmetry-constrained Ginibre $U(1)$ version robustly approaches a narrow near-marginal regime and reproduces the predicted low-frequency finite-width spectral behavior. These results support the use of symmetry-guided effective stability diagnostics as practical principles for stochastic architecture search in controlled settings.

Neural fields, also known as implicit neural representations (INRs), offer a powerful framework for modeling continuous geometry, but their effectiveness in high-dimensional scientific settings is limited by slow convergence and scaling challenges. In this study, we extend INR models to handle spatiotemporal and multivariate signals and show how INR features can be transferred across scientific signals to enable efficient and scalable representation across time and ensemble runs in an amortized fashion. Across controlled transformation regimes (e.g., geometric transformations and localized perturbations of synthetic fields) and high-fidelity scientific domains-including turbulent flows, fluid-material impact dynamics, and astrophysical systems-we show that transferable features improve not only signal fidelity but also the accuracy of derived geometric and physical quantities, including density gradients and vorticity. In particular, transferable features reduce iterations to reach target reconstruction quality by up to an order of magnitude, increase early-stage reconstruction quality by multiple dB (with gains exceeding 10 dB in some cases), and consistently improve gradient-based physical accuracy.

The multi-scale and non-linear nature of phase-field models of solidification requires fine spatial and temporal discretization, leading to long computation times. This could be overcome with artificial-intelligence approaches. Surrogate models based on neural operators could have a lower computational cost than conventional numerical discretization methods. We propose a new neural operator approach that bridges classical convex-concave splitting schemes with physics-informed learning to accelerate the simulation of phase-field models. It consists of a Deep Ritz method, where a neural operator is trained to approximate a variational formulation of the phase-field model. By training the neural operator with an energy-splitting variational formulation, we enforce the energy dissipation property of the underlying models. We further introduce a custom Reaction-Diffusion Neural Operator (RDNO) architecture, adapted to the operators of the model equations. We successfully apply the deep learning approach to the isotropic Allen-Cahn equation and to anisotropic dendritic growth simulation. We demonstrate that our physically-informed training provides better generalization in out-of-distribution evaluations than data-driven training, while achieving faster inference than traditional Fourier spectral methods.

Reconstructing continuous environmental fields from sparse and irregular observations remains a central challenge in environmental modelling and biodiversity informatics. Many ecological datasets are heterogeneous in space and time, making grid-based approaches difficult to scale or generalise across domains. Here, we evaluate implicit neural representations (INRs) as a coordinate-based modelling framework for learning continuous spatial and spatio-temporal fields directly from coordinate inputs. We analyse their behaviour across three representative modelling scenarios: species distribution reconstruction, phenological dynamics, and morphological segmentation derived from open biodiversity data. Beyond predictive performance, we examine interpolation behaviour, spatial coherence, and computational characteristics relevant for environmental modelling workflows, including scalability, resolution-independent querying, and architectural inductive bias. Results show that neural fields provide stable continuous representations with predictable computational cost, complementing classical smoothers and tree-based approaches. These findings position coordinate-based neural fields as a flexible representation layer that can be integrated into environmental modelling pipelines and exploratory analysis frameworks for large, irregularly sampled datasets.

Inverse rendering is an ill-posed problem, but priors such as illumination priors can help simplify it. Existing work either disregards the spherical and rotation-equivariant nature of illumination environments or does not provide a well-behaved latent space. We propose a rotation-equivariant variational autoencoder that models natural illumination on the sphere without relying on 2D projections. To preserve the SO(2)-equivariance of environment maps, we use a novel Vector Neuron Vision Transformer (VN-ViT) as encoder and a rotation-equivariant conditional neural field as decoder. In the encoder, we reduce the equivariance from SO(3) to SO(2) using a novel SO(2)-equivariant fully connected layer, an extension of Vector Neurons. We show that our SO(2)-equivariant fully connected layer outperforms standard Vector Neurons when used in our SO(2)-equivariant model. Compared to previous methods, our variational autoencoder enables smoother interpolation in latent space and offers a more well-behaved latent space.

We investigate the potential of weights to serve as effective representations, focusing on neural fields. Our key insight is that constraining the optimization space through a pre-trained base model and low-rank adaptation (LoRA) can induce structure in weight space. Across reconstruction, generation, and analysis tasks on 2D and 3D data, we find that multiplicative LoRA weights achieve high representation quality while exhibiting distinctiveness and semantic structure. When used with latent diffusion models, multiplicative LoRA weights enable higher-quality generation than existing weight-space methods.