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.
135 papers in Foundation models · newest first
cs.LG
Neural PDE solvers have followed the scaling trajectory of vision and language, with recent foundation models reaching billions of parameters. We argue that scale is a poor substitute for architectural inductive bias in this domain: structured priors deliver outsized parameter efficiency, and the pattern of where they succeed and fail is itself informative about what they capture. We instantiate this argument in WaveLiT, an architecture combining a discrete wavelet transform for lossless multi-resolution tokenization, an augmented linear attention block, a shared-weight multiscale feature pyramid, and a wavelet-domain auxiliary loss. Bespoke 1-10M-parameter WaveLiT models compete with foundation models of 100-1000 their size across eight TheWell benchmarks, with the largest gains on wave and acoustic-dominated benchmarks where the wavelet-multiscale prior fits the dominant dynamical structure and small per-step errors do not compound geometrically under rollout. Trained jointly across all eight benchmarks, a 10M-parameter foundation variant exhibits a structured, physically interpretable transfer pattern – strongest where the wavelet-multiscale prior matches the dynamics, weakest on chaotic advection-dominated flows. The entire pipeline trains on a single GPU. The results suggest that small-model PDE performance is shaped by architectural inductive bias rather than scale, and that the structure of a prior's failures is a useful empirical signal about its content.
cs.CE
Data-driven thermal predictors for 3D-ICs are often trained from scratch for each chip design using many high-fidelity finite-element simulations, leading to high data-generation cost and costly cross-design reuse. We propose Therm-FM, a neural operator framework that adapts a pretrained partial differential equation (PDE) foundation model to steady-state and transient 3D-IC thermal simulation. The motivation is that steady-state and transient chip-level heat conduction respectively share elliptic and parabolic operator structures with diffusion-type PDEs, allowing pretrained diffusion priors to provide an effective initialization for thermal-field prediction under heterogeneous materials, dense TSV/microbump interconnects, and package-level boundary conditions. To further reduce data-generation cost, Therm-FM incorporates a thermal-equivalent multi-fidelity training strategy that uses low-cost approximate simulations for thermal-domain adaptation and limited high-fidelity samples for calibration. Experiments on public HotSpot benchmarks and industrial 3D-IC package benchmarks show that Therm-FM achieves up to a 10.6x reduction in mean error and surpasses prior best accuracy with less than 20% of the training data. In cross-chip adaptation, it matches or surpasses full-data baselines in several metrics using only 10–30 target samples. We release datasets, source code, and pretrained models at https://github.com/haiyangxin/Therm-FM.
cs.CL
Large Language Models (LLMs) have become the predominant paradigm in NLP, advancing both research and industry. As model sizes and pretraining data grow, concerns about Pretraining Data Exposure (PDE) increase due to the scale and opacity of training datasets. PDE refers to determining whether specific data appeared in an LLM's pretraining corpus. It is critical for ensuring evaluation integrity and protecting privacy, intersecting two key areas: data contamination and membership inference. Though conceptually related, these areas have often been studied in isolation. This paper offers the first unified survey of both under the PDE framework. We formalize PDE across exposure levels, review attack and defense methods, synthesize empirical findings, and highlight open challenges and future research directions.
cs.LG
Partial differential equation (PDE) foundation models are pretrained networks that forecast how physical fields like velocity and pressure evolve from a single reusable solver. On unfamiliar flows their predictions drift step by step, errors concentrate in a few regions, yet retraining destabilizes the network and uniform post-hoc correction overlooks this spatial concentration. To address this, we propose a frozen-solver post-hoc correction framework, Adaptive Risk-Calibrated Spatial Triage for Auditable Refinement (ARC-STAR). ARC-STAR organizes correction into three stages: a global corrector removes broad solver bias, a blockwise local refiner cleans the post-global residual, and, at deployment, a label-free score routes refinement to high-risk blocks under a compute budget. The framework is designed to be (i) frozen-host, preserving the pretrained solver without fine-tuning; (ii) auditable, with global and local stages trained and evaluated separately for measurable contributions; and (iii) budget-aware, using a blockwise interface that either refines the full field or routes limited compute to high-risk regions. Across five flow benchmarks spanning ten regime cells, ARC-STAR is the only method that cuts velocity rollout error by at least 36x over raw Poseidon on every cell. The global stage reduces raw host error by 91-99%, and the local stage further reduces the remaining post-global residual by up to 94.4%.
cs.LG
Pre-training neural operators on diverse partial differential equation (PDE) datasets has emerged as a promising direction for building general-purpose surrogate models in scientific machine learning. However, the inherent complexity and structural diversity of PDE solution operators make multi-PDE pre-training fundamentally challenging. Existing methods mainly address this by increasing model capacity, while leaving the target solution operators unchanged. Inspired by classical numerical analysis, we instead propose to transform complex and diverse solution operators into simpler, better-aligned forms that are easier to model jointly. Since the optimal transformation varies across PDE types, it must be adaptive and input-dependent, allowing a single neural operator to approximate an entire family of operators. We instantiate this idea as AOT-POT (adaptive operator-transformation for pre-training operator transformer), which expands hidden representations into multiple parallel streams, adaptively aggregates and redistributes them before and after each sub-layer, and mixes streams through Sinkhorn-projected doubly stochastic matrices for stable training. These mechanisms together reshape diverse solution operators into a unified form that can be effectively modeled by a single architecture. Empirically, AOT-POT achieves state-of-the-art performance on 12 PDE benchmarks with only 3% additional parameters, reducing relative L2 error by up to 77.6% (40.9% on average). Fine-tuning AOT-POT further reduces L2 error by up to 92% on in-domain PDEs and 89% on out-of-domain PDEs (unseen types during pre-training), demonstrating that adaptive operator transformation is an effective and complementary direction for advancing PDE foundation models beyond simply scaling model capacity.
cs.LG
We introduce Tadpole, a novel foundation model for three-dimensional partial differential equations (PDEs) that addresses key challenges in transferability, scalability to high dimensionality, and multi-functionality. Tadpole is pre-trained as an autoencoder on synthetic 3D PDE data generated by an efficient online data-generation framework. This enables large-scale, diverse training without storage or I/O overhead, demonstrated by scaling to an equivalent of hundreds of terabytes of training data. By autoencoding single-channel spatial crops, Tadpole learns rich and transferable representations across heterogeneous physical systems with varying numbers of state variables and spatial resolutions. Although pre-trained solely as an autoencoder, Tadpole can be efficiently applied for multiple downstream tasks beyond reconstruction, including dynamics learning and generative modeling. For dynamics learning, we propose a novel parameter-efficient fine-tuning strategy that integrates low-rank adaptation, latent-space transformations, and reintroduced skip connections, achieving accurate temporal modeling with a minimal number of trainable parameters. Tadpole demonstrates strong fine-tuning performance across various downstream tasks, highlighting its versatility and effectiveness as a foundation model for 3D PDE learning. Source code and pre-trained weights of Tadpole are available at https://tum-pbs.github.io/tadpole/.
cs.LG
Scaling Scientific Machine Learning (SciML) toward universal foundation models is bottlenecked by negative transfer: the simultaneous co-training of disparate partial differential equation (PDE) regimes can induce gradient conflict, unstable optimization, and plasticity loss in dense neural operators. In particular, broadband open-channel fluid dynamics and boundary-dominated porous media flows impose incompatible spectral and geometric demands on a single dense parameter path. We introduce Shodh-MoE, a sparse-activated latent transformer architecture for multi-physics transport. Shodh-MoE operates on compressed 16^3 physical latents produced by a physics-informed autoencoder with an intra-tokenizer Helmholtz-style velocity parameterization, restricting decoded states to divergence-free velocity manifolds. The model guarantees exact mass conservation, achieving a physically verifiable velocity divergence of ~2.8 x 10^-10 (evaluated post-hoc in FP64) on 128^3 grids. A Top-1 soft-semantic router dynamically assigns localized latent patches to expert subnetworks, enabling specialized parameter paths for distinct physical mechanisms while preserving shared experts for universal symmetries. In a 20,000-step distributed pretraining run over mixed three-dimensional physical tensors, routing telemetry shows autonomous domain bifurcation: held-out validation tokens from the open-channel domain route exclusively to Expert 0, while porous-media tokens route exclusively to Expert 1. The model converges simultaneously across both regimes, achieving latent validation MSEs of 2.46 x 10^-5 and 9.76 x 10^-6, and decoded physical MSEs of 2.48 x 10^-6 and 1.76 x 10^-6. These results support sparse expert routing as a practical architectural mechanism for mitigating multi-physics interference in universal neural operators.
cs.LG
Partial differential equations (PDEs) govern diverse physical phenomena, yet high-fidelity numerical solutions are computationally expensive and Machine Learning approaches lack generalization. While Scientific Foundation Models (SFMs) aim to provide universal surrogates, typical encoding-decoding approaches suffer from high pretraining costs and limited interpretability. In this paper, we propose Compositional Neural Operators (CompNO) for 2D systems, a framework that decomposes complex PDEs into a library of Foundation Blocks. Each block is a specialized Neural Operator pretrained on elementary physics. This modular library contains convection, diffusion, and nonlinear convection blocks as well as a Poisson Solver, enabling the framework to address the pressure-velocity coupling. These experts are assembled via an Adaptation Block featuring an Aggregator. This aggregator learns nonlinear interactions by minimizing data loss and physics-based residuals driven from governing equations. The proposed approach has been evaluated on the Convection-Diffusion equation, the Burgers' equation, and the Incompressible Navier-Stokes equation. Our results demonstrate that learning from elementary operators significantly improves adaptability, enhances model interpretability and facilitates the reuse of pretrained blocks when adapting to new physical systems.
cs.LG
jNO (jax Neural Operators) is a JAX-native library for neural operators and foundation models with unified support for both data-driven and physics-informed training. Its core design is a tracing system in which domains, model calls, residuals, supervised losses, and diagnostics are written in one symbolic language and compiled into one optimization pipeline. This allows users to move between operator regression, mesh-aware residual evaluation, and PDE-constrained training without restructuring the surrounding code. jNO also supports multi-model compositions, fine-grained control at parameter level (model, optimizer, and learning rate), hyperparameter tuning, and JAX-native workflows for translated PDE foundation-model families. The source repository is available at https://github.com/FhG-IISB/jNO.
cs.DC
Universal Machine Learning Interatomic Potentials (uMLIPs), pre-trained on massively diverse datasets encompassing inorganic materials and organic molecules across the entire periodic table, serve as foundational models for quantum-accurate physical simulations. However, uMLIP training requires second-order derivatives, which lack corresponding parallel training frameworks; moreover, scaling to the billion-parameter regime causes explosive growth in computation and communication overhead, making its training a tremendous challenge. We introduce MatRIS-MoE, a billion-parameter Mixture-of-Experts model built upon invariant architecture, and {Janus}, a pioneering high-dimensional distributed training framework for uMLIPs with hardware-aware optimizations. Deployed across two Exascale supercomputers, our code attains a peak performance of 1.2/1.0 EFLOPS (24%/{35.5%} of theoretical peak) in single precision at over 90% parallel efficiency, compressing the training of billion-parameter uMLIPs from weeks to hours. This work establishes a new high-water mark for AI-for-Science (AI4S) foundation models at Exascale and provides essential infrastructure for rapid scientific discovery.
cs.CV
We present PhysInOne, a large-scale synthetic dataset addressing the critical scarcity of physically-grounded training data for AI systems. Unlike existing datasets limited to merely hundreds or thousands of examples, PhysInOne provides 2 million videos across 153,810 dynamic 3D scenes, covering 71 basic physical phenomena in mechanics, optics, fluid dynamics, and magnetism. Distinct from previous works, our scenes feature multiobject interactions against complex backgrounds, with comprehensive ground-truth annotations including 3D geometry, semantics, dynamic motion, physical properties, and text descriptions. We demonstrate PhysInOne's efficacy across four emerging applications: physics-aware video generation, long-/short-term future frame prediction, physical property estimation, and motion transfer. Experiments show that fine-tuning foundation models on PhysInOne significantly enhances physical plausibility, while also exposing critical gaps in modeling complex physical dynamics and estimating intrinsic properties. As the largest dataset of its kind, orders of magnitude beyond prior works, PhysInOne establishes a new benchmark for advancing physics-grounded world models in generation, simulation, and embodied AI.
cs.LG
Multiple operator learning concerns learning operator families indexed by an operator descriptor . Training data are collected hierarchically by sampling operator instances , then input functions per instance, and finally evaluation points per input, yielding noisy observations of . While recent work has developed expressive multi-task and multiple operator learning architectures and approximation-theoretic scaling laws, quantitative statistical generalization guarantees remain limited. We provide a covering-number-based generalization analysis for separable models, focusing on the Multiple Neural Operator (MNO) architecture: we first derive explicit metric-entropy bounds for hypothesis classes given by linear combinations of products of deep ReLU subnetworks, and then combine these complexity bounds with approximation guarantees for MNO to obtain an explicit approximation-estimation tradeoff for the expected test error on new (unseen) triples . The resulting bound makes the dependence on the hierarchical sampling budgets transparent and yields an explicit learning-rate statement in the operator-sampling budget , providing a sample-complexity characterization for generalization across operator instances. The structure and architecture can also be viewed as a general purpose solver or an example of a "small'' PDE foundation model, where the triples are one form of multi-modality.
cs.LG
Reservoir simulation workflows face a fundamental data asymmetry: input parameter fields (geostatistical permeability realizations, porosity distributions) are free to generate in arbitrary quantities, yet existing neural operator surrogates require large corpora of expensive labeled simulation trajectories and cannot exploit this unlabeled structure. We introduce \textbf{PI-JEPA} (Physics-Informed Joint Embedding Predictive Architecture), a surrogate pretraining framework that trains \emph{without any completed PDE solves}, using masked latent prediction on unlabeled parameter fields under per-sub-operator PDE residual regularization. The predictor bank is structurally aligned with the Lie–Trotter operator-splitting decomposition of the governing equations, dedicating a separate physics-constrained latent module to each sub-process (pressure, saturation transport, reaction), enabling fine-tuning with as few as 100 labeled simulation runs. On single-phase Darcy flow, PI-JEPA achieves lower error than FNO and lower error than DeepONet at , with 24% improvement over supervised-only training at , demonstrating that label-free surrogate pretraining substantially reduces the simulation budget required for multiphysics surrogate deployment.
cs.LG
We investigate the generalization capabilities of In-Context Operator Networks (ICONs), a new class of operator networks that build on the principles of in-context learning, for higher-order partial differential equations. We extend previous work by expanding the type and scope of differential equations handled by the foundation model. We demonstrate that while processing complex inputs requires some new computational methods, the underlying machine learning techniques are largely consistent with simpler cases. Our implementation shows that although point-wise accuracy degrades for higher-order problems like the heat equation, the model retains qualitative accuracy in capturing solution dynamics and overall behavior. This demonstrates the model's ability to extrapolate fundamental solution characteristics to problems outside its training regime.
cs.LG
Designing urban spaces that provide pedestrian wind comfort and safety requires time-resolved Computational Fluid Dynamics (CFD) simulations, but their current computational cost makes extensive design exploration impractical. We introduce WinDiNet (Wind Diffusion Network), a pretrained video diffusion model that is repurposed as a fast, differentiable surrogate for this task. Starting from LTX-Video, a 2B-parameter latent video transformer, we fine-tune on 10,000 2D incompressible CFD simulations over procedurally generated building layouts. A systematic study of training regimes, conditioning mechanisms, and VAE adaptation strategies, including a physics-informed decoder loss, identifies a configuration that outperforms purpose-built neural PDE solvers. The resulting model generates full 112-frame rollouts in under a second. As the surrogate is end-to-end differentiable, it doubles as a physics simulator for gradient-based inverse optimization: given an urban footprint layout, we optimize building positions directly through backpropagation to improve wind safety as well as pedestrian wind comfort. Experiments on single- and multi-inlet layouts show that the optimizer discovers effective layouts even under challenging multi-objective configurations, with all improvements confirmed by ground-truth CFD simulations.
cs.LG
Partial differential equation (PDE) simulation holds extensive significance in scientific research. Currently, the integration of deep neural networks to learn solution operators of PDEs has introduced great potential. In this paper, we present UniFluids, a conditional flow-matching framework that harnesses the scalability of diffusion Transformer to unify learning of solution operators across diverse PDEs with varying dimensionality and physical variables. Unlike the autoregressive PDE foundation models, UniFluids adopts flow-matching to achieve parallel sequence generation, making it the first such approach for unified operator learning. Specifically, the introduction of a unified four-dimensional spatiotemporal representation for the heterogeneous PDE datasets enables joint training and conditional encoding. Furthermore, we find the effective dimension of the PDE dataset is much lower than its patch dimension. We thus employ -prediction in the flow-matching operator learning, which is verified to significantly improve prediction accuracy. We conduct a large-scale evaluation of UniFluids on several PDE datasets covering spatial dimensions 1D, 2D and 3D. Experimental results show that UniFluids achieves strong prediction accuracy and demonstrates good scalability and cross-scenario generalization capability. The code will be released later.
cs.RO
Robot learning requires adaptation methods that improve reliably from limited, mixed-quality interaction data. This is especially challenging in long-horizon, contact-rich tasks, where end-to-end policy finetuning remains inefficient and brittle. World models offer a compelling alternative: by predicting the outcomes of candidate action sequences, they enable online planning through counterfactual reasoning. However, training action-conditioned robotic world models directly in the real world requires diverse data at impractical scale. We introduce Simulation Distillation (SimDist), a framework that uses physics simulators as a scalable source of action-conditioned robot experience. During pretraining, SimDist distills structural priors from the simulator into a world model that enables planning from raw real-world observations. During real-world adaptation, SimDist transfers the encoder, reward model, and value function learned in simulation, and updates only the latent dynamics model using real-world prediction losses. This reduces adaptation to supervised system identification while preserving dense, long-horizon planning signals for online improvement. Across contact-rich manipulation and quadruped locomotion tasks, SimDist rapidly improves with experience, while prior adaptation methods struggle to make progress or degrade during online finetuning. Project website and code: https://sim-dist.github.io
cs.LG
Foundation models for partial differential equations (PDEs) have emerged as powerful surrogates pre-trained on diverse physical systems, but adapting them to new downstream tasks remains challenging due to limited task-specific data and distribution shifts. While fine-tuning has proven transformative in natural language processing, best practices for adapting PDE foundation models remain underexplored. Although physics-informed training has successfully trained accurate solvers across a wide range of PDE problems, its potential for fine-tuning data-based foundation models has not been systematically studied. In this work, we introduce a physics-informed fine-tuning framework that adapts pre-trained PDE foundation models by incorporating physical constraints (PDE residuals and boundary conditions) directly into the fine-tuning objective. This enables effective adaptation in data-scarce regimes while promoting physical consistency. We evaluate our method on a downstream task composed of an unseen PDE class and compare it with data-driven finetuning counterparts. Our results demonstrate that physics-informed fine-tuning achieves competitive accuracy without requiring PDE solutions for training. Furthermore, a hybrid fine-tuning strategy yields superior generalization to out-of-distribution scenarios when only minimal training data is available. These findings establish physics-informed fine-tuning as a scalable and data-efficient paradigm, providing a physically interpretable pathway for adapting foundation models in scientific machine learning.
cs.LG
Partial differential equations (PDEs) govern physical phenomena across the full range of scientific scales, yet their computational solution remains one of the defining challenges of modern science. This critical review examines two mature but epistemologically distinct paradigms for PDE solution, classical numerical methods and machine learning approaches, through a unified evaluative framework organized around six fundamental computational challenges. Classical methods are assessed for their structure-preserving properties, rigorous convergence theory, and scalable solver design; their persistent limitations in high-dimensional and geometrically complex settings are characterized precisely. Machine learning approaches are introduced under a taxonomy organized by the degree to which physical knowledge is incorporated and subjected to the same critical evaluation applied to classical methods. Classical methods are deductive – errors are bounded by quantities derivable from PDE structure and discretization parameters – while machine learning methods are inductive – accuracy depends on statistical proximity to the training distribution. This epistemological distinction is the primary criterion governing responsible method selection. We identify three genuine complementarities between the paradigms and develop principles for hybrid design, including a framework for the structure inheritance problem that addresses when classical guarantees propagate through hybrid couplings, and an error budget decomposition that separates discretization, neural approximation, and coupling contributions. We further assess emerging frontiers, including foundation models, differentiable programming, quantum algorithms, and exascale co-design, evaluating each against the structural constraints that determine whether current barriers are fundamental or contingent on engineering progress.
cs.LG
PDE foundation models are typically pretrained on large, diverse corpora of PDE datasets and can be adapted to new settings with limited task-specific data. However, most downstream evaluations focus on forward problems, such as autoregressive rollout prediction. In this work, we study an inverse problem in inertial confinement fusion (ICF): estimating system parameters (inputs) from multi-modal, snapshot-style observations (outputs). Using the open JAG benchmark, which provides hyperspectral X-ray images and scalar observables per simulation, we finetune the PDE foundation model and train a lightweight task-specific head to jointly reconstruct hyperspectral images and regress system parameters. The fine-tuned model achieves accurate hyperspectral reconstruction (test MSE 1.2e-3) and strong parameter-estimation performance (up to R^2=0.995). Data-scaling experiments (5%-100% of the training set) show consistent improvements in both reconstruction and regression losses as the amount of training data increases, with the largest marginal gains in the low-data regime. Finally, finetuning from pretrained MORPH weights outperforms training the same architecture from scratch, demonstrating that foundation-model initialization improves sample efficiency for data-limited inverse problems in ICF.