Miles Cranmer

Michael McCabe, Bruno Régaldo-Saint Blancard, Liam Holden Parker et al. · cambridge

We introduce multiple physics pretraining (MPP), an autoregressive task-agnostic pretraining approach for physical surrogate modeling of spatiotemporal systems with transformers. In MPP, rather than training one model on a specific physical system, we train a backbone model to predict the dynamics of multiple heterogeneous physical systems simultaneously in order to learn features that are broadly useful across systems and facilitate transfer. In order to learn effectively in this setting, we introduce a shared embedding and normalization strategy that projects the fields of multiple systems into a shared embedding space. We validate the efficacy of our approach on both pretraining and downstream tasks over a broad fluid mechanics-oriented benchmark. We show that a single MPP-pretrained transformer is able to match or outperform task-specific baselines on all pretraining sub-tasks without the need for finetuning. For downstream tasks, we demonstrate that finetuning MPP-trained models results in more accurate predictions across multiple time-steps on systems with previously unseen physical components or higher dimensional systems compared to training from scratch or finetuning pretrained video foundation models. We open-source our code and model weights trained at multiple scales for reproducibility.

Digvijay Wadekar, Leander Thiele, J. Colin Hill et al. · cambridge

Feedback from active galactic nuclei (AGN) and supernovae can affect measurements of integrated SZ flux of halos ($Y_\mathrm{SZ}$) from CMB surveys, and cause its relation with the halo mass ($Y_\mathrm{SZ}-M$) to deviate from the self-similar power-law prediction of the virial theorem. We perform a comprehensive study of such deviations using CAMELS, a suite of hydrodynamic simulations with extensive variations in feedback prescriptions. We use a combination of two machine learning tools (random forest and symbolic regression) to search for analogues of the $Y-M$ relation which are more robust to feedback processes for low masses ($M\lesssim 10^{14}\, h^{-1} \, M_\odot$); we find that simply replacing $Y\rightarrow Y(1+M_*/M_\mathrm{gas})$ in the relation makes it remarkably self-similar. This could serve as a robust multiwavelength mass proxy for low-mass clusters and galaxy groups. Our methodology can also be generally useful to improve the domain of validity of other astrophysical scaling relations. We also forecast that measurements of the $Y-M$ relation could provide percent-level constraints on certain combinations of feedback parameters and/or rule out a major part of the parameter space of supernova and AGN feedback models used in current state-of-the-art hydrodynamic simulations. Our results can be useful for using upcoming SZ surveys (e.g., SO, CMB-S4) and galaxy surveys (e.g., DESI and Rubin) to constrain the nature of baryonic feedback. Finally, we find that the an alternative relation, $Y-M_*$, provides complementary information on feedback than $Y-M$

Joe Watson, Chen Song, Oliver Weeger et al. · cambridge

This survey examines the broad suite of methods and models for combining machine learning with physics knowledge for prediction and forecast, with a focus on partial differential equations. These methods have attracted significant interest due to their potential impact on advancing scientific research and industrial practices by improving predictive models with small- or large-scale datasets and expressive predictive models with useful inductive biases. The survey has two parts. The first considers incorporating physics knowledge on an architectural level through objective functions, structured predictive models, and data augmentation. The second considers data as physics knowledge, which motivates looking at multi-task, meta, and contextual learning as an alternative approach to incorporating physics knowledge in a data-driven fashion. Finally, we also provide an industrial perspective on the application of these methods and a survey of the open-source ecosystem for physics-informed machine learning.

Francisco Villaescusa-Navarro, Boris Bolliet, Pablo Villanueva-Domingo et al.

We present Denario, an AI multi-agent system designed to serve as a scientific research assistant. Denario can perform many different tasks, such as generating ideas, checking the literature, developing research plans, writing and executing code, making plots, and drafting and reviewing a scientific paper. The system has a modular architecture, allowing it to handle specific tasks, such as generating an idea, or carrying out end-to-end scientific analysis using Cmbagent as a deep-research backend. In this work, we describe in detail Denario and its modules, and illustrate its capabilities by presenting multiple AI-generated papers generated by it in many different scientific disciplines such as astrophysics, biology, biophysics, biomedical informatics, chemistry, material science, mathematical physics, medicine, neuroscience and planetary science. Denario also excels at combining ideas from different disciplines, and we illustrate this by showing a paper that applies methods from quantum physics and machine learning to astrophysical data. We report the evaluations performed on these papers by domain experts, who provided both numerical scores and review-like feedback. We then highlight the strengths, weaknesses, and limitations of the current system. Finally, we discuss the ethical implications of AI-driven research and reflect on how such technology relates to the philosophy of science. We publicly release the code at https://github.com/AstroPilot-AI/Denario. A Denario demo can also be run directly on the web at https://huggingface.co/spaces/astropilot-ai/Denario, and the full app will be deployed on the cloud.

Christian Kragh Jespersen, Miles Cranmer, Peter Melchior et al. · cambridge, princeton

Efficiently mapping baryonic properties onto dark matter is a major challenge in astrophysics. Although semi-analytic models (SAMs) and hydrodynamical simulations have made impressive advances in reproducing galaxy observables across cosmologically significant volumes, these methods still require significant computation times, representing a barrier to many applications. Graph Neural Networks (GNNs) have recently proven to be the natural choice for learning physical relations. Among the most inherently graph-like structures found in astrophysics are the dark matter merger trees that encode the evolution of dark matter halos. In this paper we introduce a new, graph-based emulator framework, $\texttt{Mangrove}$, and show that it emulates the galactic stellar mass, cold gas mass and metallicity, instantaneous and time-averaged star formation rate, and black hole mass -- as predicted by a SAM -- with root mean squared error up to two times lower than other methods across a $(75 Mpc/h)^3$ simulation box in 40 seconds, 4 orders of magnitude faster than the SAM. We show that $\texttt{Mangrove}$ allows for quantification of the dependence of galaxy properties on merger history. We compare our results to the current state of the art in the field and show significant improvements for all target properties. $\texttt{Mangrove}$ is publicly available.

Pablo Lemos, Miles Cranmer, Muntazir Abidi et al. · cambridge

Simulation-based inference (SBI) is rapidly establishing itself as a standard machine learning technique for analyzing data in cosmological surveys. Despite continual improvements to the quality of density estimation by learned models, applications of such techniques to real data are entirely reliant on the generalization power of neural networks far outside the training distribution, which is mostly unconstrained. Due to the imperfections in scientist-created simulations, and the large computational expense of generating all possible parameter combinations, SBI methods in cosmology are vulnerable to such generalization issues. Here, we discuss the effects of both issues, and show how using a Bayesian neural network framework for training SBI can mitigate biases, and result in more reliable inference outside the training set. We introduce cosmoSWAG, the first application of Stochastic Weight Averaging to cosmology, and apply it to SBI trained for inference on the cosmic microwave background.

Ji Won Park, Simon Birrer, Madison Ueland et al. · cambridge

We present a Bayesian graph neural network (BGNN) that can estimate the weak lensing convergence ($κ$) from photometric measurements of galaxies along a given line of sight. The method is of particular interest in strong gravitational time delay cosmography (TDC), where characterizing the "external convergence" ($κ_{\rm ext}$) from the lens environment and line of sight is necessary for precise inference of the Hubble constant ($H_0$). Starting from a large-scale simulation with a $κ$ resolution of $\sim$1$'$, we introduce fluctuations on galaxy-galaxy lensing scales of $\sim$1$''$ and extract random sightlines to train our BGNN. We then evaluate the model on test sets with varying degrees of overlap with the training distribution. For each test set of 1,000 sightlines, the BGNN infers the individual $κ$ posteriors, which we combine in a hierarchical Bayesian model to yield constraints on the hyperparameters governing the population. For a test field well sampled by the training set, the BGNN recovers the population mean of $κ$ precisely and without bias, resulting in a contribution to the $H_0$ error budget well under 1\%. In the tails of the training set with sparse samples, the BGNN, which can ingest all available information about each sightline, extracts more $κ$ signal compared to a simplified version of the traditional method based on matching galaxy number counts, which is limited by sample variance. Our hierarchical inference pipeline using BGNNs promises to improve the $κ_{\rm ext}$ characterization for precision TDC. The implementation of our pipeline is available as a public Python package, Node to Joy.

Siavash Golkar, Mariel Pettee, Michael Eickenberg et al. · cambridge

Due in part to their discontinuous and discrete default encodings for numbers, Large Language Models (LLMs) have not yet been commonly used to process numerically-dense scientific datasets. Rendering datasets as text, however, could help aggregate diverse and multi-modal scientific data into a single training corpus, thereby potentially facilitating the development of foundation models for science. In this work, we introduce xVal, a strategy for continuously tokenizing numbers within language models that results in a more appropriate inductive bias for scientific applications. By training specially-modified language models from scratch on a variety of scientific datasets formatted as text, we find that xVal generally outperforms other common numerical tokenization strategies on metrics including out-of-distribution generalization and computational efficiency.

Thomas Pfeil, Miles Cranmer, Shirley Ho et al. · cambridge

Planet formation is a multi-scale process in which the coagulation of $\mathrm{μm}$-sized dust grains in protoplanetary disks is strongly influenced by the hydrodynamic processes on scales of astronomical units ($\approx 1.5\times 10^8 \,\mathrm{km}$). Studies are therefore dependent on subgrid models to emulate the micro physics of dust coagulation on top of a large scale hydrodynamic simulation. Numerical simulations which include the relevant physical effects are complex and computationally expensive. Here, we present a fast and accurate learned effective model for dust coagulation, trained on data from high resolution numerical coagulation simulations. Our model captures details of the dust coagulation process that were so far not tractable with other dust coagulation prescriptions with similar computational efficiency.

Payel Mukhopadhyay, Stefan S. Nixon, Romain Watteaux et al.

Whether physics foundation models can be usefully deployed on laboratory experiments remains an open question for scientific machine learning (ML). We test this question on the Rayleigh-Taylor instability (RTI), a ubiquitous and demanding fluid instability seen from tabletop flows to supernova explosions, in which small perturbations at a density interface grow into chaotic, multiscale mixing as a lighter fluid accelerates into a heavier one. Standard ML models struggle with RTI, and despite over a century of theoretical, numerical, and experimental work, it carries an unresolved discrepancy between simulation and experiment: the late-time mixing growth rate, $α$, measured in most laboratory experiments ($\sim$ 0.06-0.07), is roughly three times the value from idealized direct numerical simulations (DNS, $\sim$ 0.02). The gap's origin remains debated. These properties make RTI a stringent test for a question that matters well beyond RTI: can foundation models trained only on simulations generalise to sparse, messy, and noisy laboratory settings? We finetune Walrus, a foundation model for continuum dynamics, on three or fewer DNS realizations and recover key RTI physics over long rollouts. Applied zero-shot to sliding-barrier laboratory data, the finetuned model leaves the DNS-like regime and enters the observed growth band, having never seen a single experimental sample. These results provide independent, data-driven evidence that initial conditions play a crucial role in the longstanding sim-experiment gap in $α$. The model also generalises zero-shot to stable stratification, a buoyancy regime absent from training, correctly slowing mixing-layer growth. Together, our results show that foundation models can generalise well beyond their training data, predicting laboratory behavior and unseen physical regimes, opening new ways to probe longstanding simulation-experiment gaps.

Arya Grayeli, Atharva Sehgal, Omar Costilla-Reyes et al.

We present a novel method for symbolic regression (SR), the task of searching for compact programmatic hypotheses that best explain a dataset. The problem is commonly solved using genetic algorithms; we show that we can enhance such methods by inducing a library of abstract textual concepts. Our algorithm, called LaSR, uses zero-shot queries to a large language model (LLM) to discover and evolve concepts occurring in known high-performing hypotheses. We discover new hypotheses using a mix of standard evolutionary steps and LLM-guided steps (obtained through zero-shot LLM queries) conditioned on discovered concepts. Once discovered, hypotheses are used in a new round of concept abstraction and evolution. We validate LaSR on the Feynman equations, a popular SR benchmark, as well as a set of synthetic tasks. On these benchmarks, LaSR substantially outperforms a variety of state-of-the-art SR approaches based on deep learning and evolutionary algorithms. Moreover, we show that LaSR can be used to discover a novel and powerful scaling law for LLMs.

David Ruhe, Kaze Wong, Miles Cranmer et al. · cambridge

We propose parameterizing the population distribution of the gravitational wave population modeling framework (Hierarchical Bayesian Analysis) with a normalizing flow. We first demonstrate the merit of this method on illustrative experiments and then analyze four parameters of the latest LIGO/Virgo data release: primary mass, secondary mass, redshift, and effective spin. Our results show that despite the small and notoriously noisy dataset, the posterior predictive distributions (assuming a prior over the parameters of the flow) of the observed gravitational wave population recover structure that agrees with robust previous phenomenological modeling results while being less susceptible to biases introduced by less flexible models. Therefore, the method forms a promising flexible, reliable replacement for population inference distributions, even when data is highly noisy.

Kaze W. K Wong, Miles Cranmer · cambridge

We present an automatic approach to discover analytic population models for gravitational-wave (GW) events from data. As more gravitational-wave (GW) events are detected, flexible models such as Gaussian Mixture Models have become more important in fitting the distribution of GW properties due to their expressivity. However, flexible models come with many parameters that lack physical motivation, making interpreting the implication of these models challenging. In this work, we demonstrate symbolic regression can complement flexible models by distilling the posterior predictive distribution of such flexible models into interpretable analytic expressions. We recover common GW population models such as a power-law-plus-Gaussian, and find a new empirical population model which combines accuracy and simplicity. This demonstrates a strategy to automatically discover interpretable population models in the ever-growing GW catalog, which can potentially be applied to other astrophysical phenomena.

Christian Pedersen, Tiberiu Tesileanu, Tinghui Wu et al. · cambridge

In Elmarakeby et al., "Biologically informed deep neural network for prostate cancer discovery", a feedforward neural network with biologically informed, sparse connections (P-NET) was presented to model the state of prostate cancer. We verified the reproducibility of the study conducted by Elmarakeby et al., using both their original codebase, and our own re-implementation using more up-to-date libraries. We quantified the contribution of network sparsification by Reactome biological pathways, and confirmed its importance to P-NET's superior performance. Furthermore, we explored alternative neural architectures and approaches to incorporating biological information into the networks. We experimented with three types of graph neural networks on the same training data, and investigated the clinical prediction agreement between different models. Our analyses demonstrated that deep neural networks with distinct architectures make incorrect predictions for individual patient that are persistent across different initializations of a specific neural architecture. This suggests that different neural architectures are sensitive to different aspects of the data, an important yet under-explored challenge for clinical prediction tasks.

Ameya Daigavane, Arthur Kosmala, Miles Cranmer et al. · cambridge

Machine learning has become increasingly popular for efficiently modelling the dynamics of complex physical systems, demonstrating a capability to learn effective models for dynamics which ignore redundant degrees of freedom. Learned simulators typically predict the evolution of the system in a step-by-step manner with numerical integration techniques. However, such models often suffer from instability over long roll-outs due to the accumulation of both estimation and integration error at each prediction step. Here, we propose an alternative construction for learned physical simulators that are inspired by the concept of action-angle coordinates from classical mechanics for describing integrable systems. We propose Action-Angle Networks, which learn a nonlinear transformation from input coordinates to the action-angle space, where evolution of the system is linear. Unlike traditional learned simulators, Action-Angle Networks do not employ any higher-order numerical integration methods, making them extremely efficient at modelling the dynamics of integrable physical systems.

Caleb Lammers, Miles Cranmer, Sam Hadden et al. · cambridge

Constraining planet formation models based on the observed exoplanet population requires generating large samples of synthetic planetary systems, which can be computationally prohibitive. A significant bottleneck is simulating the giant impact phase, during which planetary embryos evolve gravitationally and combine to form planets, which may themselves experience later collisions. To accelerate giant impact simulations, we present a machine learning (ML) approach to predicting collisional outcomes in multiplanet systems. Trained on more than 500,000 $N$-body simulations of three-planet systems, we develop an ML model that can accurately predict which two planets will experience a collision, along with the state of the post-collision planets, from a short integration of the system's initial conditions. Our model greatly improves on non-ML baselines that rely on metrics from dynamics theory, which struggle to accurately predict which pair of planets will experience a collision. By combining with a model for predicting long-term stability, we create an ML-based giant impact emulator, which can predict the outcomes of giant impact simulations with reasonable accuracy and a speedup of up to four orders of magnitude. We expect our model to enable analyses that would not otherwise be computationally feasible. As such, we release our training code, along with an easy-to-use API for our collision outcome model and giant impact emulator.

Ruben Ohana, Michael McCabe, Lucas Meyer et al. · cambridge

Machine learning based surrogate models offer researchers powerful tools for accelerating simulation-based workflows. However, as standard datasets in this space often cover small classes of physical behavior, it can be difficult to evaluate the efficacy of new approaches. To address this gap, we introduce the Well: a large-scale collection of datasets containing numerical simulations of a wide variety of spatiotemporal physical systems. The Well draws from domain experts and numerical software developers to provide 15TB of data across 16 datasets covering diverse domains such as biological systems, fluid dynamics, acoustic scattering, as well as magneto-hydrodynamic simulations of extra-galactic fluids or supernova explosions. These datasets can be used individually or as part of a broader benchmark suite. To facilitate usage of the Well, we provide a unified PyTorch interface for training and evaluating models. We demonstrate the function of this library by introducing example baselines that highlight the new challenges posed by the complex dynamics of the Well. The code and data is available at https://github.com/PolymathicAI/the_well.

Philippe Martin Wyder, Judah Goldfeder, Alexey Yermakov et al.

Machine learning (ML) is transforming modeling and control in the physical, engineering, and biological sciences. However, rapid development has outpaced the creation of standardized, objective benchmarks - leading to weak baselines, reporting bias, and inconsistent evaluations across methods. This undermines reproducibility, misguides resource allocation, and obscures scientific progress. To address this, we propose a Common Task Framework (CTF) for scientific machine learning. The CTF features a curated set of datasets and task-specific metrics spanning forecasting, state reconstruction, and generalization under realistic constraints, including noise and limited data. Inspired by the success of CTFs in fields like natural language processing and computer vision, our framework provides a structured, rigorous foundation for head-to-head evaluation of diverse algorithms. As a first step, we benchmark methods on two canonical nonlinear systems: Kuramoto-Sivashinsky and Lorenz. These results illustrate the utility of the CTF in revealing method strengths, limitations, and suitability for specific classes of problems and diverse objectives. Next, we are launching a competition around a global real world sea surface temperature dataset with a true holdout dataset to foster community engagement. Our long-term vision is to replace ad hoc comparisons with standardized evaluations on hidden test sets that raise the bar for rigor and reproducibility in scientific ML.

Stefano Riva, Carolina Introini, Antonio Cammi et al.

The demand for clean energy is ever increasing, with new nuclear technologies presenting a complementary solution to renewable energies. However, designing and operating these systems is exceptionally difficult, given the complexity of the physical phenomena that interact to form the system dynamics. While high-fidelity simulations help to understand the non-linear, multi-physics interactions within a reactor, they are computationally expensive and rarely suitable for real-time applications. Furthermore, model-based approaches are inherently sensitive to simplifying assumptions required to derive their governing equations and parameters, leading to inevitable discrepancies with real-world measurements. In contrast, Machine Learning (ML) methods have the potential to generate reliable surrogate models which may be able to quickly predict the system's behaviour. However, the number of data-driven methods that can potentially be used for this task is large and diverse. In a safety-critical setting such as nuclear engineering, a fair comparison of different ML methods, and a clear understanding of their advantages and limitations, is of paramount importance. To address this, we introduce a Common Task Framework (CTF) for ML in nuclear engineering, building upon previous efforts in dynamical systems and seismology. This CTF considers a curated set of datasets from different nuclear and nuclear-adjacent systems. The CTF evaluates the performance of a method on 12 established metrics, alongside a new paradigm focused on system monitoring from sparse measurements only. We illustrate the framework by benchmarking standard ML baselines against these datasets, revealing current method limitations. Our vision is to replace ad hoc comparisons with standardized evaluations on hidden test sets, raising the bar for rigour and reproducibility in scientific ML for the nuclear industry.

Hadi Sotoudeh, Payel Mukhopadhyay, Ruben Ohana et al.

We investigate the impact of tokeniser pretraining on the accuracy and efficiency of physics emulation. Modern high-resolution simulations produce vast volumes of data spanning diverse physical regimes and scales. Training foundation models to learn the dynamics underlying such data enables the modelling of complex multiphysics phenomena, especially in data-limited settings. The emerging class of physics foundation models typically aims to learn two tasks jointly: (i) extracting compact representations of high-resolution spatiotemporal data, and (ii) capturing governing physical dynamics. However, learning both tasks from scratch simultaneously can impede the effectiveness of either process. We show that pretraining the tokeniser with an autoencoding objective prior to training the dynamics model enhances computational efficiency for physics emulation. Notably, the magnitude of this benefit depends on domain alignment: pretraining on the same physical system as the emulation task yields the largest improvements, while pretraining on other systems provides moderate gains. In-domain pretraining reduces VRMSE by 64% after 10,500 training steps compared to training from scratch. To our knowledge, this is the first systematic investigation of tokeniser pretraining for physics foundation models. We further introduce flexible spatiotemporal compression operations that extend causal convolutions to support runtime-adjustable compression ratios, enabling efficient adaptation to diverse downstream tasks. Our findings provide practical guidance for training efficient physics emulators and highlight the importance of strategic pretraining data selection.

Licong Xu, Milind Sarkar, Anto I. Lonappan et al.

We present a multi-agent system for automation of scientific research tasks, cmbagent (https://github.com/CMBAgents/cmbagent). The system is formed by about 30 Large Language Model (LLM) agents and implements a Planning & Control strategy to orchestrate the agentic workflow, with no human-in-the-loop at any point. Each agent specializes in a different task (performing retrieval on scientific papers and codebases, writing code, interpreting results, critiquing the output of other agents) and the system is able to execute code locally. We successfully apply cmbagent to carry out a PhD level cosmology task (the measurement of cosmological parameters using supernova data) and evaluate its performance on two benchmark sets, finding superior performance over state-of-the-art LLMs. The source code is available on GitHub, demonstration videos are also available, and the system is deployed on HuggingFace and will be available on the cloud.

Cristiana Diaconu, Miles Cranmer, Richard E. Turner et al.

Dominant approaches for modelling Partial Differential Equations (PDEs) rely on deterministic predictions, yet many physical systems of interest are inherently chaotic and uncertain. While training probabilistic models from scratch is possible, it is computationally expensive and fails to leverage the significant resources already invested in high-performing deterministic backbones. In this work, we adopt a training-efficient strategy to transform pre-trained deterministic models into probabilistic ones via retrofitting with a proper scoring rule: the Continuous Ranked Probability Score (CRPS). Crucially, this approach is architecture-agnostic: it applies the same adaptation mechanism across distinct model backbones with minimal code modifications. The method proves highly effective across different scales of pre-training: for models trained on single dynamical systems, we achieve 20-54% reductions in rollout CRPS and up to 30% improvements in variance-normalised RMSE (VRMSE) relative to compute-matched deterministic fine-tuning. We further validate our approach on a PDE foundation model, trained on multiple systems and retrofitted on the dataset of interest, to show that our probabilistic adaptation yields an improvement of up to 40% in CRPS and up to 15% in VRMSE compared to deterministic fine-tuning. Validated across diverse architectures and dynamics, our results show that probabilistic PDE modelling need not require retraining from scratch, but can be unlocked from existing deterministic backbones with modest additional training cost.

Alexey Yermakov, Yue Zhao, Marine Denolle et al.

Seismology faces fundamental challenges in state forecasting and reconstruction (e.g., earthquake early warning and ground motion prediction) and managing the parametric variability of source locations, mechanisms, and Earth models (e.g., subsurface structure and topography effects). Addressing these with simulations is hindered by their massive scale, both in synthetic data volumes and numerical complexity, while real-data efforts are constrained by models that inadequately reflect the Earth's complexity and by sparse sensor measurements from the field. Recent machine learning (ML) efforts offer promise, but progress is obscured by a lack of proper characterization, fair reporting, and rigorous comparisons. To address this, we introduce a Common Task Framework (CTF) for ML for seismic wavefields, starting with three distinct wavefield datasets. Our CTF features a curated set of datasets at various scales (global, crustal, and local) and task-specific metrics spanning forecasting, reconstruction, and generalization under realistic constraints such as noise and limited data. Inspired by CTFs in fields like natural language processing, this framework provides a structured and rigorous foundation for head-to-head algorithm evaluation. We illustrate the evaluation procedure with scores reported for two of the datasets, showcasing the performance of various methods and foundation models for reconstructing seismic wavefields from both simulated and real-world sensor measurements. The CTF scores reveal the strengths, limitations, and suitability for specific problem classes. Our vision is to replace ad hoc comparisons with standardized evaluations on hidden test sets, raising the bar for rigor and reproducibility in scientific ML.

Miles Cranmer

PySR is an open-source library for practical symbolic regression, a type of machine learning which aims to discover human-interpretable symbolic models. PySR was developed to democratize and popularize symbolic regression for the sciences, and is built on a high-performance distributed back-end, a flexible search algorithm, and interfaces with several deep learning packages. PySR's internal search algorithm is a multi-population evolutionary algorithm, which consists of a unique evolve-simplify-optimize loop, designed for optimization of unknown scalar constants in newly-discovered empirical expressions. PySR's backend is the extremely optimized Julia library SymbolicRegression.jl, which can be used directly from Julia. It is capable of fusing user-defined operators into SIMD kernels at runtime, performing automatic differentiation, and distributing populations of expressions to thousands of cores across a cluster. In describing this software, we also introduce a new benchmark, "EmpiricalBench," to quantify the applicability of symbolic regression algorithms in science. This benchmark measures recovery of historical empirical equations from original and synthetic datasets.

Miles Cranmer, Daniel Tamayo, Hanno Rein et al.

Despite over three hundred years of effort, no solutions exist for predicting when a general planetary configuration will become unstable. We introduce a deep learning architecture to push forward this problem for compact systems. While current machine learning algorithms in this area rely on scientist-derived instability metrics, our new technique learns its own metrics from scratch, enabled by a novel internal structure inspired from dynamics theory. Our Bayesian neural network model can accurately predict not only if, but also when a compact planetary system with three or more planets will go unstable. Our model, trained directly from short N-body time series of raw orbital elements, is more than two orders of magnitude more accurate at predicting instability times than analytical estimators, while also reducing the bias of existing machine learning algorithms by nearly a factor of three. Despite being trained on compact resonant and near-resonant three-planet configurations, the model demonstrates robust generalization to both non-resonant and higher multiplicity configurations, in the latter case outperforming models fit to that specific set of integrations. The model computes instability estimates up to five orders of magnitude faster than a numerical integrator, and unlike previous efforts provides confidence intervals on its predictions. Our inference model is publicly available in the SPOCK package, with training code open-sourced.

Elizabeth S. Z. Tan, Adil Soubki, Miles Cranmer

Symbolic distillation replaces neural networks, or components thereof, with interpretable, closed-form mathematical expressions. This approach has shown promise in discovering physical laws and mathematical relationships directly from trained deep learning models, yet adoption remains limited due to the engineering barrier of integrating symbolic regression into deep learning workflows. We introduce SymTorch, a library that automates this distillation by wrapping neural network components, collecting their input-output behavior, and approximating them with human-readable equations via PySR. SymTorch handles the engineering challenges that have hindered adoption: GPU-CPU data transfer, input-output caching, model serialization, and seamless switching between neural and symbolic forward passes. We demonstrate SymTorch across diverse architectures including GNNs, PINNs and transformer models. Finally, we present a proof-of-concept for accelerating LLM inference by replacing MLP layers with symbolic surrogates, achieving an 8.3\% throughput improvement with moderate performance degradation.

Siavash Golkar, Jake Kovalic, Irina Espejo Morales et al.

Biological function emerges from coupled constraints across sequence, structure, regulation, evolution, and cellular context, yet most foundation models in biology are trained within one modality or for a fixed forward task. We present MIMIC, a generative multimodal foundation model trained on our newly curated and aligned dataset, LORE, linking nucleic acid, protein, evolutionary, structural, regulatory, and semantic/contextual modalities within partially observed biomolecular states. MIMIC uses a split-track encoder-decoder architecture to condition on arbitrary subsets of observed modalities and reconstruct or generate missing components of molecular state across the genome, transcriptome, and proteome. Multimodal conditioning consistently improves MIMIC's sequence reconstruction relative to sequence-only inputs, while its learned representations enable state-of-the-art performance on RNA and protein downstream tasks. MIMIC achieves state-of-the-art splicing prediction, and its joint generative formulation enables isoform-aware inference that further improves performance. Beyond prediction, the same generative framework supports constrained design. For RNA, MIMIC identifies corrective edits in a clinically relevant HBB splice-disrupting mutation without reverting it by using evolutionary and structural signals. For proteins, jointly conditioning on shape and surface chemistry of PD-L1 and hACE2 binding sites produces diverse, high-confidence sequences with strong in silico support for target binding. Finally, MIMIC uses experimental context as semantic conditioning to model assay-dependent RNA chemical probing, rather than treating context as a fixed output. Together, these results position MIMIC's aligned multimodal generative modeling as a strong foundation for unifying representation learning, conditional prediction, and constrained biomolecular design within a single model.

Ho Fung Tsoi, Dylan Rankin, Cecile Caillol et al.

We introduce SymbolFit, a framework that automates parametric modeling by using symbolic regression to perform a machine-search for functions that fit the data while simultaneously providing uncertainty estimates in a single run. Traditionally, constructing a parametric model to accurately describe binned data has been a manual and iterative process, requiring an adequate functional form to be determined before the fit can be performed. The main challenge arises when the appropriate functional forms cannot be derived from first principles, especially when there is no underlying true closed-form function for the distribution. In this work, we develop a framework that automates and streamlines the process by utilizing symbolic regression, a machine learning technique that explores a vast space of candidate functions without requiring a predefined functional form because the functional form itself is treated as a trainable parameter, making the process far more efficient and effortless than traditional regression methods. We demonstrate the framework in high-energy physics experiments at the CERN Large Hadron Collider (LHC) using five real proton-proton collision datasets from new physics searches, including background modeling in resonance searches for high-mass dijet, trijet, paired-dijet, diphoton, and dimuon events. We show that our framework can flexibly and efficiently generate a wide range of candidate functions that fit a nontrivial distribution well using a simple fit configuration that varies only by random seed, and that the same fit configuration, which defines a vast function space, can also be applied to distributions of different shapes, whereas achieving a comparable result with traditional methods would have required extensive manual effort.

Guilherme S. Imai Aldeia, Hengzhe Zhang, Geoffrey Bomarito et al.

Symbolic Regression (SR) is a powerful technique for discovering interpretable mathematical expressions. However, benchmarking SR methods remains challenging due to the diversity of algorithms, datasets, and evaluation criteria. In this work, we present an updated version of SRBench. Our benchmark expands the previous one by nearly doubling the number of evaluated methods, refining evaluation metrics, and using improved visualizations of the results to understand the performances. Additionally, we analyze trade-offs between model complexity, accuracy, and energy consumption. Our results show that no single algorithm dominates across all datasets. We propose a call for action from SR community in maintaining and evolving SRBench as a living benchmark that reflects the state-of-the-art in symbolic regression, by standardizing hyperparameter tuning, execution constraints, and computational resource allocation. We also propose deprecation criteria to maintain the benchmark's relevance and discuss best practices for improving SR algorithms, such as adaptive hyperparameter tuning and energy-efficient implementations.

Payel Mukhopadhyay, Michael McCabe, Ruben Ohana et al. · cambridge

Patch-based transformer surrogates have become increasingly effective for modeling spatiotemporal dynamics, but the fixed patch size is a major limitation for budget-conscience deployment in production. We introduce two lightweight, architecture-agnostic modules-the Convolutional Kernel Modulator (CKM) and Convolutional Stride Modulator (CSM)-that enable dynamic patch size control at inference in patch based models, without retraining or accuracy loss. Combined with a cyclic patch-size rollout, our method mitigates patch artifacts and improves long-term stability for video-like prediction tasks. Applied to a range of challenging 2D and 3D PDE benchmarks, our approach improves rollout fidelity and runtime efficiency. To our knowledge, this is the first framework to enable inference-time patch-size tunability in patch-based PDE surrogates. Its plug-and-play design makes it broadly applicable across architectures-establishing a general foundation for compute-adaptive modeling in PDE surrogate tasks.

Michael McCabe, Payel Mukhopadhyay, Tanya Marwah et al. · cambridge

Foundation models have transformed machine learning for language and vision, but achieving comparable impact in physical simulation remains a challenge. Data heterogeneity and unstable long-term dynamics inhibit learning from sufficiently diverse dynamics, while varying resolutions and dimensionalities challenge efficient training on modern hardware. Through empirical and theoretical analysis, we incorporate new approaches to mitigate these obstacles, including a harmonic-analysis-based stabilization method, load-balanced distributed 2D and 3D training strategies, and compute-adaptive tokenization. Using these tools, we develop Walrus, a transformer-based foundation model developed primarily for fluid-like continuum dynamics. Walrus is pretrained on nineteen diverse scenarios spanning astrophysics, geoscience, rheology, plasma physics, acoustics, and classical fluids. Experiments show that Walrus outperforms prior foundation models on both short and long term prediction horizons on downstream tasks and across the breadth of pretraining data, while ablation studies confirm the value of our contributions to forecast stability, training throughput, and transfer performance over conventional approaches. Code and weights are released for community use.

Rio Alexa Fear, Payel Mukhopadhyay, Michael McCabe et al.

Recent advances in mechanistic interpretability have revealed that large language models (LLMs) develop internal representations corresponding not only to concrete entities but also distinct, human-understandable abstract concepts and behaviour. Moreover, these hidden features can be directly manipulated to steer model behaviour. However, it remains an open question whether this phenomenon is unique to models trained on inherently structured data (ie. language, images) or if it is a general property of foundation models. In this work, we investigate the internal representations of a large physics-focused foundation model. Inspired by recent work identifying single directions in activation space for complex behaviours in LLMs, we extract activation vectors from the model during forward passes over simulation datasets for different physical regimes. We then compute "delta" representations between the two regimes. These delta tensors act as concept directions in activation space, encoding specific physical features. By injecting these concept directions back into the model during inference, we can steer its predictions, demonstrating causal control over physical behaviours, such as inducing or removing some particular physical feature from a simulation. These results suggest that scientific foundation models learn generalised representations of physical principles. They do not merely rely on superficial correlations and patterns in the simulations. Our findings open new avenues for understanding and controlling scientific foundation models and has implications for AI-enabled scientific discovery.

Rudy Morel, Francesco Pio Ramunno, Jeff Shen et al.

Conditional diffusion models provide a natural framework for probabilistic prediction of dynamical systems and have been successfully applied to fluid dynamics and weather prediction. However, in many settings, the available information at a given time represents only a small fraction of what is needed to predict future states, either due to measurement uncertainty or because only a small fraction of the state can be observed. This is true for example in solar physics, where we can observe the Sun's surface and atmosphere, but its evolution is driven by internal processes for which we lack direct measurements. In this paper, we tackle the probabilistic prediction of partially observable, long-memory dynamical systems, with applications to solar dynamics and the evolution of active regions. We show that standard inference schemes, such as autoregressive rollouts, fail to capture long-range dependencies in the data, largely because they do not integrate past information effectively. To overcome this, we propose a multiscale inference scheme for diffusion models, tailored to physical processes. Our method generates trajectories that are temporally fine-grained near the present and coarser as we move farther away, which enables capturing long-range temporal dependencies without increasing computational cost. When integrated into a diffusion model, we show that our inference scheme significantly reduces the bias of the predicted distributions and improves rollout stability.

Jeff Shen, Francois Lanusse, Liam Holden Parker et al. · cambridge

Sequential scientific data span many resolutions and domains, and unifying them into a common representation is a key step toward developing foundation models for the sciences. Astronomical spectra exemplify this challenge: massive surveys have collected millions of spectra across a wide range of wavelengths and resolutions, yet analyses remain fragmented across spectral domains (e.g., optical vs. infrared) and object types (e.g., stars vs. galaxies), limiting the ability to pool information across datasets. We present a deep learning model that jointly learns from heterogeneous spectra in a self-supervised manner. Our universal spectral tokenizer processes spectra from a variety of object types and resolutions directly on their native wavelength grids, producing intrinsically aligned, homogeneous, and physically meaningful representations that can be efficiently adapted to achieve competitive performance across a range of downstream tasks. For the first time, we demonstrate that a single model can unify spectral data across resolutions and domains, suggesting that our model can serve as a powerful building block for foundation models in astronomy -- and potentially extend to other scientific domains with heterogeneous sequential data, such as climate and healthcare.

Liam Parker, Francois Lanusse, Siavash Golkar et al.

We present AstroCLIP, a single, versatile model that can embed both galaxy images and spectra into a shared, physically meaningful latent space. These embeddings can then be used - without any model fine-tuning - for a variety of downstream tasks including (1) accurate in-modality and cross-modality semantic similarity search, (2) photometric redshift estimation, (3) galaxy property estimation from both images and spectra, and (4) morphology classification. Our approach to implementing AstroCLIP consists of two parts. First, we embed galaxy images and spectra separately by pretraining separate transformer-based image and spectrum encoders in self-supervised settings. We then align the encoders using a contrastive loss. We apply our method to spectra from the Dark Energy Spectroscopic Instrument and images from its corresponding Legacy Imaging Survey. Overall, we find remarkable performance on all downstream tasks, even relative to supervised baselines. For example, for a task like photometric redshift prediction, we find similar performance to a specifically-trained ResNet18, and for additional tasks like physical property estimation (stellar mass, age, metallicity, and sSFR), we beat this supervised baseline by 19\% in terms of $R^2$. We also compare our results to a state-of-the-art self-supervised single-modal model for galaxy images, and find that our approach outperforms this benchmark by roughly a factor of two on photometric redshift estimation and physical property prediction in terms of $R^2$, while remaining roughly in-line in terms of morphology classification. Ultimately, our approach represents the first cross-modal self-supervised model for galaxies, and the first self-supervised transformer-based architectures for galaxy images and spectra.

Ho Fung Tsoi, Adrian Alan Pol, Vladimir Loncar et al.

The high-energy physics community is investigating the potential of deploying machine-learning-based solutions on Field-Programmable Gate Arrays (FPGAs) to enhance physics sensitivity while still meeting data processing time constraints. In this contribution, we introduce a novel end-to-end procedure that utilizes a machine learning technique called symbolic regression (SR). It searches the equation space to discover algebraic relations approximating a dataset. We use PySR (a software to uncover these expressions based on an evolutionary algorithm) and extend the functionality of hls4ml (a package for machine learning inference in FPGAs) to support PySR-generated expressions for resource-constrained production environments. Deep learning models often optimize the top metric by pinning the network size because the vast hyperparameter space prevents an extensive search for neural architecture. Conversely, SR selects a set of models on the Pareto front, which allows for optimizing the performance-resource trade-off directly. By embedding symbolic forms, our implementation can dramatically reduce the computational resources needed to perform critical tasks. We validate our method on a physics benchmark: the multiclass classification of jets produced in simulated proton-proton collisions at the CERN Large Hadron Collider. We show that our approach can approximate a 3-layer neural network using an inference model that achieves up to a 13-fold decrease in execution time, down to 5 ns, while still preserving more than 90% approximation accuracy.

Leander Thiele, Miles Cranmer, William Coulton et al.

Theoretical uncertainty limits our ability to extract cosmological information from baryonic fields such as the thermal Sunyaev-Zel'dovich (tSZ) effect. Being sourced by the electron pressure field, the tSZ effect depends on baryonic physics that is usually modeled by expensive hydrodynamic simulations. We train neural networks on the IllustrisTNG-300 cosmological simulation to predict the continuous electron pressure field in galaxy clusters from gravity-only simulations. Modeling clusters is challenging for neural networks as most of the gas pressure is concentrated in a handful of voxels and even the largest hydrodynamical simulations contain only a few hundred clusters that can be used for training. Instead of conventional convolutional neural net (CNN) architectures, we choose to employ a rotationally equivariant DeepSets architecture to operate directly on the set of dark matter particles. We argue that set-based architectures provide distinct advantages over CNNs. For example, we can enforce exact rotational and permutation equivariance, incorporate existing knowledge on the tSZ field, and work with sparse fields as are standard in cosmology. We compose our architecture with separate, physically meaningful modules, making it amenable to interpretation. For example, we can separately study the influence of local and cluster-scale environment, determine that cluster triaxiality has negligible impact, and train a module that corrects for mis-centering. Our model improves by 70 % on analytic profiles fit to the same simulation data. We argue that the electron pressure field, viewed as a function of a gravity-only simulation, has inherent stochasticity, and model this property through a conditional-VAE extension to the network. This modification yields further improvement by 7 %, it is limited by our small training set however. (abridged)

Pablo Lemos, Niall Jeffrey, Miles Cranmer et al.

We present an approach for using machine learning to automatically discover the governing equations and hidden properties of real physical systems from observations. We train a "graph neural network" to simulate the dynamics of our solar system's Sun, planets, and large moons from 30 years of trajectory data. We then use symbolic regression to discover an analytical expression for the force law implicitly learned by the neural network, which our results showed is equivalent to Newton's law of gravitation. The key assumptions that were required were translational and rotational equivariance, and Newton's second and third laws of motion. Our approach correctly discovered the form of the symbolic force law. Furthermore, our approach did not require any assumptions about the masses of planets and moons or physical constants. They, too, were accurately inferred through our methods. Though, of course, the classical law of gravitation has been known since Isaac Newton, our result serves as a validation that our method can discover unknown laws and hidden properties from observed data. More broadly this work represents a key step toward realizing the potential of machine learning for accelerating scientific discovery.

Digvijay Wadekar, Leander Thiele, Francisco Villaescusa-Navarro et al.

Complex astrophysical systems often exhibit low-scatter relations between observable properties (e.g., luminosity, velocity dispersion, oscillation period). These scaling relations illuminate the underlying physics, and can provide observational tools for estimating masses and distances. Machine learning can provide a fast and systematic way to search for new scaling relations (or for simple extensions to existing relations) in abstract high-dimensional parameter spaces. We use a machine learning tool called symbolic regression (SR), which models patterns in a dataset in the form of analytic equations. We focus on the Sunyaev-Zeldovich flux$-$cluster mass relation ($Y_\mathrm{SZ}-M$), the scatter in which affects inference of cosmological parameters from cluster abundance data. Using SR on the data from the IllustrisTNG hydrodynamical simulation, we find a new proxy for cluster mass which combines $Y_\mathrm{SZ}$ and concentration of ionized gas ($c_\mathrm{gas}$): $M \propto Y_\mathrm{conc}^{3/5} \equiv Y_\mathrm{SZ}^{3/5} (1-A\, c_\mathrm{gas})$. $Y_\mathrm{conc}$ reduces the scatter in the predicted $M$ by $\sim 20-30$\% for large clusters ($M\gtrsim 10^{14}\, h^{-1} \, M_\odot$), as compared to using just $Y_\mathrm{SZ}$. We show that the dependence on $c_\mathrm{gas}$ is linked to cores of clusters exhibiting larger scatter than their outskirts. Finally, we test $Y_\mathrm{conc}$ on clusters from CAMELS simulations and show that $Y_\mathrm{conc}$ is robust against variations in cosmology, subgrid physics, and cosmic variance. Our results and methodology can be useful for accurate multiwavelength cluster mass estimation from upcoming CMB and X-ray surveys like ACT, SO, eROSITA and CMB-S4.

Kimberly Stachenfeld, Drummond B. Fielding, Dmitrii Kochkov et al.

Turbulence simulation with classical numerical solvers requires high-resolution grids to accurately resolve dynamics. Here we train learned simulators at low spatial and temporal resolutions to capture turbulent dynamics generated at high resolution. We show that our proposed model can simulate turbulent dynamics more accurately than classical numerical solvers at the comparably low resolutions across various scientifically relevant metrics. Our model is trained end-to-end from data and is capable of learning a range of challenging chaotic and turbulent dynamics at low resolution, including trajectories generated by the state-of-the-art Athena++ engine. We show that our simpler, general-purpose architecture outperforms various more specialized, turbulence-specific architectures from the learned turbulence simulation literature. In general, we see that learned simulators yield unstable trajectories; however, we show that tuning training noise and temporal downsampling solves this problem. We also find that while generalization beyond the training distribution is a challenge for learned models, training noise, added loss constraints, and dataset augmentation can help. Broadly, we conclude that our learned simulator outperforms traditional solvers run on coarser grids, and emphasize that simple design choices can offer stability and robust generalization.

Miles Cranmer, Peter Melchior, Brian Nord

We present an approach for maximizing a global utility function by learning how to allocate resources in an unsupervised way. We expect interactions between allocation targets to be important and therefore propose to learn the reward structure for near-optimal allocation policies with a GNN. By relaxing the resource constraint, we can employ gradient-based optimization in contrast to more standard evolutionary algorithms. Our algorithm is motivated by a problem in modern astronomy, where one needs to select-based on limited initial information-among $10^9$ galaxies those whose detailed measurement will lead to optimal inference of the composition of the universe. Our technique presents a way of flexibly learning an allocation strategy by only requiring forward simulators for the physics of interest and the measurement process. We anticipate that our technique will also find applications in a range of resource allocation problems.

V. Ashley Villar, Miles Cranmer, Edo Berger et al.

There is a shortage of multi-wavelength and spectroscopic followup capabilities given the number of transient and variable astrophysical events discovered through wide-field, optical surveys such as the upcoming Vera C. Rubin Observatory. From the haystack of potential science targets, astronomers must allocate scarce resources to study a selection of needles in real time. Here we present a variational recurrent autoencoder neural network to encode simulated Rubin Observatory extragalactic transient events using 1% of the PLAsTiCC dataset to train the autoencoder. Our unsupervised method uniquely works with unlabeled, real time, multivariate and aperiodic data. We rank 1,129,184 events based on an anomaly score estimated using an isolation forest. We find that our pipeline successfully ranks rarer classes of transients as more anomalous. Using simple cuts in anomaly score and uncertainty, we identify a pure (~95% pure) sample of rare transients (i.e., transients other than Type Ia, Type II and Type Ibc supernovae) including superluminous and pair-instability supernovae. Finally, our algorithm is able to identify these transients as anomalous well before peak, enabling real-time follow up studies in the era of the Rubin Observatory.

V. Ashley Villar, Miles Cranmer, Gabriella Contardo et al.

Supernovae mark the explosive deaths of stars and enrich the cosmos with heavy elements. Future telescopes will discover thousands of new supernovae nightly, creating a need to flag astrophysically interesting events rapidly for followup study. Ideally, such an anomaly detection pipeline would be independent of our current knowledge and be sensitive to unexpected phenomena. Here we present an unsupervised method to search for anomalous time series in real time for transient, multivariate, and aperiodic signals. We use a RNN-based variational autoencoder to encode supernova time series and an isolation forest to search for anomalous events in the learned encoded space. We apply this method to a simulated dataset of 12,159 supernovae, successfully discovering anomalous supernovae and objects with catastrophically incorrect redshift measurements. This work is the first anomaly detection pipeline for supernovae which works with online datastreams.

Ademola Oladosu, Tony Xu, Philip Ekfeldt et al.

This paper presents a meta-learning framework for few-shots One-Class Classification (OCC) at test-time, a setting where labeled examples are only available for the positive class, and no supervision is given for the negative example. We consider that we have a set of `one-class classification' objective-tasks with only a small set of positive examples available for each task, and a set of training tasks with full supervision (i.e. highly imbalanced classification). We propose an approach using order-equivariant networks to learn a 'meta' binary-classifier. The model will take as input an example to classify from a given task, as well as the corresponding supervised set of positive examples for this OCC task. Thus, the output of the model will be 'conditioned' on the available positive example of a given task, allowing to predict on new tasks and new examples without labeled negative examples. In this paper, we are motivated by an astronomy application. Our goal is to identify if stars belong to a specific stellar group (the 'one-class' for a given task), called \textit{stellar streams}, where each stellar stream is a different OCC-task. We show that our method transfers well on unseen (test) synthetic streams, and outperforms the baselines even though it is not retrained and accesses a much smaller part of the data per task to predict (only positive supervision). We see however that it doesn't transfer as well on the real stream GD-1. This could come from intrinsic differences from the synthetic and real stream, highlighting the need for consistency in the 'nature' of the task for this method. However, light fine-tuning improve performances and outperform our baselines. Our experiments show encouraging results to further explore meta-learning methods for OCC tasks.

Miles Cranmer, Alvaro Sanchez-Gonzalez, Peter Battaglia et al.

We develop a general approach to distill symbolic representations of a learned deep model by introducing strong inductive biases. We focus on Graph Neural Networks (GNNs). The technique works as follows: we first encourage sparse latent representations when we train a GNN in a supervised setting, then we apply symbolic regression to components of the learned model to extract explicit physical relations. We find the correct known equations, including force laws and Hamiltonians, can be extracted from the neural network. We then apply our method to a non-trivial cosmology example-a detailed dark matter simulation-and discover a new analytic formula which can predict the concentration of dark matter from the mass distribution of nearby cosmic structures. The symbolic expressions extracted from the GNN using our technique also generalized to out-of-distribution data better than the GNN itself. Our approach offers alternative directions for interpreting neural networks and discovering novel physical principles from the representations they learn.

Miles Cranmer, Sam Greydanus, Stephan Hoyer et al.

Accurate models of the world are built upon notions of its underlying symmetries. In physics, these symmetries correspond to conservation laws, such as for energy and momentum. Yet even though neural network models see increasing use in the physical sciences, they struggle to learn these symmetries. In this paper, we propose Lagrangian Neural Networks (LNNs), which can parameterize arbitrary Lagrangians using neural networks. In contrast to models that learn Hamiltonians, LNNs do not require canonical coordinates, and thus perform well in situations where canonical momenta are unknown or difficult to compute. Unlike previous approaches, our method does not restrict the functional form of learned energies and will produce energy-conserving models for a variety of tasks. We test our approach on a double pendulum and a relativistic particle, demonstrating energy conservation where a baseline approach incurs dissipation and modeling relativity without canonical coordinates where a Hamiltonian approach fails. Finally, we show how this model can be applied to graphs and continuous systems using a Lagrangian Graph Network, and demonstrate it on the 1D wave equation.