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Siddhartha Mishra

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22 papers

2 author rows

NeurIPS Conference 2025 Conference Paper

Geometry Aware Operator Transformer as an efficient and accurate neural surrogate for PDEs on arbitrary domains

Shizheng Wen
Arsh Kumbhat
Levi Lingsch
Sepehr Mousavi
Yizhou Zhao
Praveen Chandrashekar
Siddhartha Mishra

The very challenging task of learning solution operators of PDEs on arbitrary domains accurately and efficiently is of vital importance to engineering and industrial simulations. Despite the existence of many operator learning algorithms to approximate such PDEs, we find that accurate models are not necessarily computationally efficient and vice versa. We address this issue by proposing a geometry aware operator transformer (GAOT) for learning PDEs on arbitrary domains. GAOT combines novel multiscale attentional graph neural operator encoders and decoders, together with geometry embeddings and (vision) transformer processors to accurately map information about the domain and the inputs into a robust approximation of the PDE solution. Multiple innovations in the implementation of GAOT also ensure computational efficiency and scalability. We demonstrate this significant gain in both accuracy and efficiency of GAOT over several baselines on a large number of learning tasks from a diverse set of PDEs, including achieving state of the art performance on three large scale three-dimensional industrial CFD datasets. Our project page for accessing the source code is available at https: //camlab-ethz. github. io/GAOT.

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NeurIPS Conference 2025 Conference Paper

HyPINO: Multi-Physics Neural Operators via HyperPINNs and the Method of Manufactured Solutions

Rafael Bischof
Michal Piovarci
Michael Kraus
Siddhartha Mishra
Bernd Bickel

We present HyPINO, a multi-physics neural operator designed for zero-shot generalization across a broad class of PDEs without requiring task-specific fine-tuning. Our approach combines a Swin Transformer-based hypernetwork with mixed supervision: (i) labeled data from analytical solutions generated via the Method of Manufactured Solutions (MMS), and (ii) unlabeled samples optimized using physics-informed objectives. The model maps PDE parameterizations to target Physics-Informed Neural Networks (PINNs) and can handle linear elliptic, hyperbolic, and parabolic equations in two dimensions with varying source terms, geometries, and mixed Dirichlet/Neumann boundary conditions, including interior boundaries. HyPINO achieves strong zero-shot accuracy on seven benchmark problems from PINN literature, outperforming U-Nets, Poseidon, and Physics-Informed Neural Operators (PINO). Further, we introduce an iterative refinement procedure that treats the residual of the generated PINN as "delta PDE" and performs another forward pass to generate a corrective PINN. Summing their contributions and repeating this process forms an ensemble whose combined solution progressively reduces the error on six benchmarks and achieves a >100× lower $L_2$ loss in the best case, while retaining forward-only inference. Additionally, we evaluate the fine-tuning behavior of PINNs initialized by HyPINO and show that they converge faster and to lower final error than both randomly initialized and Reptile-meta-learned PINNs on five benchmarks, performing on par on the remaining two. Our results highlight the potential of this scalable approach as a foundation for extending neural operators toward solving increasingly complex, nonlinear, and high-dimensional PDE problems. The code and model weights are publicly available at https: //github. com/rbischof/hypino.

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NeurIPS Conference 2025 Conference Paper

RIGNO: A Graph-based Framework For Robust And Accurate Operator Learning For PDEs On Arbitrary Domains

Sepehr Mousavi
Shizheng Wen
Levi Lingsch
Maximilian Herde
Bogdan Raonic
Siddhartha Mishra

Learning the solution operators of PDEs on arbitrary domains is challenging due to the diversity of possible domain shapes, in addition to the often intricate underlying physics. We propose an end-to-end graph neural network (GNN) based neural operator to learn PDE solution operators from data on point clouds in arbitrary domains. Our multi-scale model maps data between input/output point clouds by passing it through a downsampled regional mesh. The approach includes novel elements aimed at ensuring spatio-temporal resolution invariance. Our model, termed RIGNO, is tested on a challenging suite of benchmarks composed of various time-dependent and steady PDEs defined on a diverse set of domains. We demonstrate that RIGNO is significantly more accurate than neural operator baselines and robustly generalizes to unseen resolutions both in space and in time. Our code is publicly available at github. com/camlab-ethz/rigno.

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ICLR Conference 2024 Conference Paper

An operator preconditioning perspective on training in physics-informed machine learning

Tim De Ryck
Florent Bonnet
Siddhartha Mishra
Emmanuel de Bézenac

In this paper, we investigate the behavior of gradient descent algorithms in physics-informed machine learning methods like PINNs, which minimize residuals connected to partial differential equations (PDEs). Our key result is that the difficulty in training these models is closely related to the conditioning of a specific differential operator. This operator, in turn, is associated to the Hermitian square of the differential operator of the underlying PDE. If this operator is ill-conditioned, it results in slow or infeasible training. Therefore, preconditioning this operator is crucial. We employ both rigorous mathematical analysis and empirical evaluations to investigate various strategies, explaining how they better condition this critical operator, and consequently improve training.

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ICML Conference 2024 Conference Paper

Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domains

Levi E. Lingsch
Mike Yan Michelis
Emmanuel de Bézenac
Sirani M. Perera
Robert K. Katzschmann
Siddhartha Mishra

The computational efficiency of many neural operators, widely used for learning solutions of PDEs, relies on the fast Fourier transform (FFT) for performing spectral computations. As the FFT is limited to equispaced (rectangular) grids, this limits the efficiency of such neural operators when applied to problems where the input and output functions need to be processed on general non-equispaced point distributions. Leveraging the observation that a limited set of Fourier (Spectral) modes suffice to provide the required expressivity of a neural operator, we propose a simple method, based on the efficient direct evaluation of the underlying spectral transformation, to extend neural operators to arbitrary domains. An efficient implementation of such direct spectral evaluations is coupled with existing neural operator models to allow the processing of data on arbitrary non-equispaced distributions of points. With extensive empirical evaluation, we demonstrate that the proposed method allows us to extend neural operators to arbitrary point distributions with significant gains in training speed over baselines, while retaining or improving the accuracy of Fourier neural operators (FNOs) and related neural operators.

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NeurIPS Conference 2024 Conference Paper

FUSE: Fast Unified Simulation and Estimation for PDEs

Levi E. Lingsch
Dana Grund
Siddhartha Mishra
Georgios Kissas

The joint prediction of continuous fields and statistical estimation of the underlying discrete parameters is a common problem for many physical systems, governed by PDEs. Hitherto, it has been separately addressed by employing operator learning surrogates for field prediction while using simulation-based inference (and its variants) for statistical parameter determination. Here, we argue that solving both problems within the same framework can lead to consistent gains in accuracy and robustness. To this end, we propose a novel and flexible formulation of the operator learning problem that jointly predicts continuous quantities and infers distributions of discrete parameters, thereby amortizing the cost of both the inverse and the surrogate models to a joint pre-training step. We present the capabilities of the proposed methodology for predicting continuous and discrete biomarkers in full-body haemodynamics simulations under different levels of missing information. We also consider a test case for atmospheric large-eddy simulation of a two-dimensional dry cold bubble, where we infer both continuous time-series and information about the system's conditions. We present comparisons against different baselines to showcase significantly increased accuracy in both the inverse and the surrogate tasks.

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TMLR Journal 2024 Journal Article

How does over-squashing affect the power of GNNs?

Francesco Di Giovanni
T. Konstantin Rusch
Michael Bronstein
Andreea Deac
Marc Lackenby
Siddhartha Mishra
Petar Veličković

Graph Neural Networks (GNNs) are the state-of-the-art model for machine learning on graph-structured data. The most popular class of GNNs operate by exchanging information between adjacent nodes, and are known as Message Passing Neural Networks (MPNNs). While understanding the expressive power of MPNNs is a key question, existing results typically consider settings with uninformative node features. In this paper, we provide a rigorous analysis to determine which function classes of node features can be learned by an MPNN of a given capacity. We do so by measuring the level of *pairwise interactions* between nodes that MPNNs allow for. This measure provides a novel quantitative characterization of the so-called over-squashing effect, which is observed to occur when a large volume of messages is aggregated into fixed-size vectors. Using our measure, we prove that, to guarantee sufficient communication between pairs of nodes, the capacity of the MPNN must be large enough, depending on properties of the input graph structure, such as commute times. For many relevant scenarios, our analysis results in impossibility statements in practice, showing that *over-squashing hinders the expressive power of MPNNs*. Our theory also holds for geometric graphs and hence extends to equivariant MPNNs on point clouds. We validate our analysis through extensive controlled experiments and ablation studies.

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NeurIPS Conference 2024 Conference Paper

Poseidon: Efficient Foundation Models for PDEs

Maximilian Herde
Bogdan Raonić
Tobias Rohner
Roger Käppeli
Roberto Molinaro
Emmanuel de Bézenac
Siddhartha Mishra

We introduce Poseidon, a foundation model for learning the solution operators of PDEs. It is based on a multiscale operator transformer, with time-conditioned layer norms that enable continuous-in-time evaluations. A novel training strategy leveraging the semi-group property of time-dependent PDEs to allow for significant scaling-up of the training data is also proposed. Poseidon is pretrained on a diverse, large scale dataset for the governing equations of fluid dynamics. It is then evaluated on a suite of 15 challenging downstream tasks that include a wide variety of PDE types and operators. We show that Poseidon exhibits excellent performance across the board by outperforming baselines significantly, both in terms of sample efficiency and accuracy. Poseidon also generalizes very well to new physics that is not seen during pretraining. Moreover, Poseidon scales with respect to model and data size, both for pretraining and for downstream tasks. Taken together, our results showcase the surprising ability of Poseidon to learn effective representations from a very small set of PDEs during pretraining in order to generalize well to unseen and unrelated PDEs downstream, demonstrating its potential as an effective, general purpose PDE foundation model. Finally, the Poseidon model as well as underlying pretraining and downstream datasets are open sourced, with code being available at https: //github. com/camlab-ethz/poseidon and pretrained models and datasets at https: //huggingface. co/camlab-ethz.

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NeurIPS Conference 2024 Conference Paper

SmallToLarge (S2L): Scalable Data Selection for Fine-tuning Large Language Models by Summarizing Training Trajectories of Small Models

Yu Yang
Siddhartha Mishra
Jeffrey Chiang
Baharan Mirzasoleiman

Despite the effectiveness of data selection for pretraining and instruction fine-tuninglarge language models (LLMs), improving data efficiency in supervised fine-tuning(SFT) for specialized domains poses significant challenges due to the complexityof fine-tuning data. To bridge this gap, we introduce an effective and scalabledata selection method for SFT, SmallToLarge (S2L), which trains a smallmodel, clusters loss trajectories of the examples, and samples from these clusters toguide data selection for larger models. We prove that during fine-tuning, sampleswithin the same loss trajectory cluster exhibit similar gradients. Then, we showthat S2L subsets have a bounded gradient error w. r. t. the full data, hence guaranteeconvergence to the neighborhood of the optimal solution. We demonstrate throughextensive experiments that S2L significantly improves data efficiency in SFT formathematical problem-solving, reducing the training data requirement to just $11$%of the original MathInstruct dataset to match full dataset performance whileoutperforming state-of-the-art data selection algorithms by an average of $4. 7$%across $6$ in- and out-domain evaluation datasets. Remarkably, selecting only 50Kdata for SFT, S2L achieves a $32. 7$% accuracy on the challenging MATHbenchmark, improving Phi-2 by $16. 6$%. In clinical text summarization on theMIMIC-III dataset, S2L again outperforms training on the full dataset usingonly $50$% of the data. Notably, S2L can perform scalable data selection using areference model $100\times$ smaller than the target model, proportionally reducing thecomputational cost.

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NeurIPS Conference 2023 Conference Paper

Convolutional Neural Operators for robust and accurate learning of PDEs

Bogdan Raonic
Roberto Molinaro
Tim De Ryck
Tobias Rohner
Francesca Bartolucci
Rima Alaifari
Siddhartha Mishra
Emmanuel de Bézenac

Although very successfully used in conventional machine learning, convolution based neural network architectures -- believed to be inconsistent in function space -- have been largely ignored in the context of learning solution operators of PDEs. Here, we present novel adaptations for convolutional neural networks to demonstrate that they are indeed able to process functions as inputs and outputs. The resulting architecture, termed as convolutional neural operators (CNOs), is designed specifically to preserve its underlying continuous nature, even when implemented in a discretized form on a computer. We prove a universality theorem to show that CNOs can approximate operators arising in PDEs to desired accuracy. CNOs are tested on a novel suite of benchmarks, encompassing a diverse set of PDEs with multi-scale solutions and are observed to significantly outperform baselines, paving the way for an alternative framework for robust and accurate operator learning.

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ICLR Conference 2023 Conference Paper

Gradient Gating for Deep Multi-Rate Learning on Graphs

T. Konstantin Rusch
Ben Chamberlain 0001
Michael W. Mahoney
Michael M. Bronstein
Siddhartha Mishra

We present Gradient Gating (G$^2$), a novel framework for improving the performance of Graph Neural Networks (GNNs). Our framework is based on gating the output of GNN layers with a mechanism for multi-rate flow of message passing information across nodes of the underlying graph. Local gradients are harnessed to further modulate message passing updates. Our framework flexibly allows one to use any basic GNN layer as a wrapper around which the multi-rate gradient gating mechanism is built. We rigorously prove that G$^2$ alleviates the oversmoothing problem and allows the design of deep GNNs. Empirical results are presented to demonstrate that the proposed framework achieves state-of-the-art performance on a variety of graph learning tasks, including on large-scale heterophilic graphs.

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ICML Conference 2023 Conference Paper

Neural Inverse Operators for Solving PDE Inverse Problems

Roberto Molinaro
Yunan Yang
Björn Engquist
Siddhartha Mishra

A large class of inverse problems for PDEs are only well-defined as mappings from operators to functions. Existing operator learning frameworks map functions to functions and need to be modified to learn inverse maps from data. We propose a novel architecture termed Neural Inverse Operators (NIOs) to solve these PDE inverse problems. Motivated by the underlying mathematical structure, NIO is based on a suitable composition of DeepONets and FNOs to approximate mappings from operators to functions. A variety of experiments are presented to demonstrate that NIOs significantly outperform baselines and solve PDE inverse problems robustly, accurately and are several orders of magnitude faster than existing direct and PDE-constrained optimization methods.

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NeurIPS Conference 2023 Conference Paper

Neural Oscillators are Universal

Samuel Lanthaler
T. Konstantin Rusch
Siddhartha Mishra

Coupled oscillators are being increasingly used as the basis of machine learning (ML) architectures, for instance in sequence modeling, graph representation learning and in physical neural networks that are used in analog ML devices. We introduce an abstract class of neural oscillators that encompasses these architectures and prove that neural oscillators are universal, i. e, they can approximate any continuous and casual operator mapping between time-varying functions, to desired accuracy. This universality result provides theoretical justification for the use of oscillator based ML systems. The proof builds on a fundamental result of independent interest, which shows that a combination of forced harmonic oscillators with a nonlinear read-out suffices to approximate the underlying operators.

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ICLR Conference 2023 Conference Paper

Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuities

Samuel Lanthaler
Roberto Molinaro
Patrik Hadorn
Siddhartha Mishra

Discontinuous solutions arise in a large class of hyperbolic and advection-dominated PDEs. This paper investigates, both theoretically and empirically, the operator learning of PDEs with discontinuous solutions. We rigorously prove, in terms of lower approximation bounds, that methods which entail a linear reconstruction step (e.g. DeepONets or PCA-Nets) fail to efficiently approximate the solution operator of such PDEs. In contrast, we show that certain methods employing a non-linear reconstruction mechanism can overcome these fundamental lower bounds and approximate the underlying operator efficiently. The latter class includes Fourier Neural Operators and a novel extension of DeepONets termed shift-DeepONets. Our theoretical findings are confirmed by empirical results for advection equations, inviscid Burgers’ equation and the compressible Euler equations of gas dynamics.

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NeurIPS Conference 2023 Conference Paper

Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learning

Francesca Bartolucci
Emmanuel de Bézenac
Bogdan Raonic
Roberto Molinaro
Siddhartha Mishra
Rima Alaifari

Recently, operator learning, or learning mappings between infinite-dimensional function spaces, has garnered significant attention, notably in relation to learning partial differential equations from data. Conceptually clear when outlined on paper, neural operators necessitate discretization in the transition to computer implementations. This step can compromise their integrity, often causing them to deviate from the underlying operators. This research offers a fresh take on neural operators with a framework Representation equivalent Neural Operators (ReNO) designed to address these issues. At its core is the concept of operator aliasing, which measures inconsistency between neural operators and their discrete representations. We explore this for widely-used operator learning techniques. Our findings detail how aliasing introduces errors when handling different discretizations and grids and loss of crucial continuous structures. More generally, this framework not only sheds light on existing challenges but, given its constructive and broad nature, also potentially offers tools for developing new neural operators.

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AAAI Conference 2022 Conference Paper

An Evaluative Measure of Clustering Methods Incorporating Hyperparameter Sensitivity

Siddhartha Mishra
Nicholas Monath
Michael Boratko
Ariel Kobren
Andrew McCallum

Clustering algorithms are often evaluated using metrics which compare with ground-truth cluster assignments, such as Rand index and NMI. Algorithm performance may vary widely for different hyperparameters, however, and thus model selection based on optimal performance for these metrics is discordant with how these algorithms are applied in practice, where labels are unavailable and tuning is often more art than science. It is therefore desirable to compare clustering algorithms not only on their optimally tuned performance, but also some notion of how realistic it would be to obtain this performance in practice. We propose an evaluation of clustering methods capturing this ease-of-tuning by modeling the expected best clustering score under a given computation budget. To encourage the adoption of the proposed metric alongside classic clustering evaluations, we provide an extensible benchmarking framework. We perform an extensive empirical evaluation of our proposed metric on popular clustering algorithms over a large collection of datasets from different domains, and observe that our new metric leads to several noteworthy observations.

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NeurIPS Conference 2022 Conference Paper

Generic bounds on the approximation error for physics-informed (and) operator learning

Tim De Ryck
Siddhartha Mishra

We propose a very general framework for deriving rigorous bounds on the approximation error for physics-informed neural networks (PINNs) and operator learning architectures such as DeepONets and FNOs as well as for physics-informed operator learning. These bounds guarantee that PINNs and (physics-informed) DeepONets or FNOs will efficiently approximate the underlying solution or solution-operator of generic partial differential equations (PDEs). Our framework utilizes existing neural network approximation results to obtain bounds on more-involved learning architectures for PDEs. We illustrate the general framework by deriving the first rigorous bounds on the approximation error of physics-informed operator learning and by showing that PINNs (and physics-informed DeepONets and FNOs) mitigate the curse of dimensionality in approximating nonlinear parabolic PDEs.

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ICML Conference 2022 Conference Paper

Graph-Coupled Oscillator Networks

T. Konstantin Rusch
Ben Chamberlain 0001
James Rowbottom
Siddhartha Mishra
Michael M. Bronstein

We propose Graph-Coupled Oscillator Networks (GraphCON), a novel framework for deep learning on graphs. It is based on discretizations of a second-order system of ordinary differential equations (ODEs), which model a network of nonlinear controlled and damped oscillators, coupled via the adjacency structure of the underlying graph. The flexibility of our framework permits any basic GNN layer (e. g. convolutional or attentional) as the coupling function, from which a multi-layer deep neural network is built up via the dynamics of the proposed ODEs. We relate the oversmoothing problem, commonly encountered in GNNs, to the stability of steady states of the underlying ODE and show that zero-Dirichlet energy steady states are not stable for our proposed ODEs. This demonstrates that the proposed framework mitigates the oversmoothing problem. Moreover, we prove that GraphCON mitigates the exploding and vanishing gradients problem to facilitate training of deep multi-layer GNNs. Finally, we show that our approach offers competitive performance with respect to the state-of-the-art on a variety of graph-based learning tasks.

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ICLR Conference 2022 Conference Paper

Long Expressive Memory for Sequence Modeling

T. Konstantin Rusch
Siddhartha Mishra
N. Benjamin Erichson
Michael W. Mahoney

We propose a novel method called Long Expressive Memory (LEM) for learning long-term sequential dependencies. LEM is gradient-based, it can efficiently process sequential tasks with very long-term dependencies, and it is sufficiently expressive to be able to learn complicated input-output maps. To derive LEM, we consider a system of multiscale ordinary differential equations, as well as a suitable time-discretization of this system. For LEM, we derive rigorous bounds to show the mitigation of the exploding and vanishing gradients problem, a well-known challenge for gradient-based recurrent sequential learning methods. We also prove that LEM can approximate a large class of dynamical systems to high accuracy. Our empirical results, ranging from image and time-series classification through dynamical systems prediction to speech recognition and language modeling, demonstrate that LEM outperforms state-of-the-art recurrent neural networks, gated recurrent units, and long short-term memory models.

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ICLR Conference 2021 Conference Paper

Coupled Oscillatory Recurrent Neural Network (coRNN): An accurate and (gradient) stable architecture for learning long time dependencies

T. Konstantin Rusch
Siddhartha Mishra

Circuits of biological neurons, such as in the functional parts of the brain can be modeled as networks of coupled oscillators. Inspired by the ability of these systems to express a rich set of outputs while keeping (gradients of) state variables bounded, we propose a novel architecture for recurrent neural networks. Our proposed RNN is based on a time-discretization of a system of second-order ordinary differential equations, modeling networks of controlled nonlinear oscillators. We prove precise bounds on the gradients of the hidden states, leading to the mitigation of the exploding and vanishing gradient problem for this RNN. Experiments show that the proposed RNN is comparable in performance to the state of the art on a variety of benchmarks, demonstrating the potential of this architecture to provide stable and accurate RNNs for processing complex sequential data.

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JMLR Journal 2021 Journal Article

On Universal Approximation and Error Bounds for Fourier Neural Operators

Nikola Kovachki
Samuel Lanthaler
Siddhartha Mishra

Fourier neural operators (FNOs) have recently been proposed as an effective framework for learning operators that map between infinite-dimensional spaces. We prove that FNOs are universal, in the sense that they can approximate any continuous operator to desired accuracy. Moreover, we suggest a mechanism by which FNOs can approximate operators associated with PDEs efficiently. Explicit error bounds are derived to show that the size of the FNO, approximating operators associated with a Darcy type elliptic PDE and with the incompressible Navier-Stokes equations of fluid dynamics, only increases sub (log)-linearly in terms of the reciprocal of the error. Thus, FNOs are shown to efficiently approximate operators arising in a large class of PDEs. [abs] [ pdf ][ bib ] &copy JMLR 2021. ( edit, beta )

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ICML Conference 2021 Conference Paper

UnICORNN: A recurrent model for learning very long time dependencies

T. Konstantin Rusch
Siddhartha Mishra

The design of recurrent neural networks (RNNs) to accurately process sequential inputs with long-time dependencies is very challenging on account of the exploding and vanishing gradient problem. To overcome this, we propose a novel RNN architecture which is based on a structure preserving discretization of a Hamiltonian system of second-order ordinary differential equations that models networks of oscillators. The resulting RNN is fast, invertible (in time), memory efficient and we derive rigorous bounds on the hidden state gradients to prove the mitigation of the exploding and vanishing gradient problem. A suite of experiments are presented to demonstrate that the proposed RNN provides state of the art performance on a variety of learning tasks with (very) long-time dependencies.

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