Title: Physics-informed Reduced Order Modeling of Time-dependent PDEs via Differentiable Solvers
Venue: NeurIPS 2025
Accepted: yes
Keywords: Reduced Order Modeling

Physics-informed Machine Learning

Differentiable Physics
Abstract: Reduced-order modeling (ROM) of time-dependent and parameterized differential equations aims to accelerate the simulation of complex high-dimensional systems by learning a compact latent manifold representation that captures the characteristics of the solution fields and their time-dependent dynamics. Although high-fidelity numerical solvers generate the training datasets, they have thus far been excluded from the training process, causing the learned latent dynamics to drift away from the discretized governing physics. This mismatch often limits generalization and forecasting capabilities. In this work, we propose **Ph**ysics-**i**nformed **ROM** ($\Phi$-ROM) by incorporating differentiable PDE solvers into the training procedure. Specifically, the latent space dynamics and its dependence on PDE parameters are shaped directly by the governing physics encoded in the solver, ensuring a strong correspondence between the full and reduced systems. Our model outperforms state-of-the-art data-driven ROMs and other physics-informed strategies by accurately generalizing to new dynamics arising from unseen parameters, enabling long-term forecasting beyond the training horizon, maintaining continuity in both time and space, and reducing the data cost. Furthermore, $\Phi$-ROM learns to recover and forecast the solution fields even when trained or evaluated with sparse and irregular observations of the fields, providing a flexible framework for field reconstruction and data assimilation. We demonstrate the framework’s robustness across various PDE solvers and highlight its broad applicability by providing an open-source JAX implementation that is readily extensible to other PDE systems and differentiable solvers, available at https://phi-rom.github.io.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a physics-informed framework for reduced-order modeling (ROM) of time-dependent partial differential equations (PDEs), with a focus on improving generalization and physical consistency. The authors propose Φ-ROM, a novel method that integrates differentiable PDE solvers into the learning pipeline, allowing the latent space dynamics to be directly shaped by the governing physics - this is unlike traditional data-driven ROMs, which often rely solely on high-fidelity simulations for supervision and may suffer from latent space drift. This alignment between latent and physical dynamics enables better forecasting, particularly under extrapolation to unseen initial conditions or parameter regimes. The authors demonstrate the effectiveness of their method across different PDE systems, showing improved long-term prediction accuracy, lower data requirements, and better physical interpretability.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: The paper is technically sound, with a clear motivation and rigorous formulation of the proposed Φ-ROM framework. By integrating differentiable solvers into the latent dynamics, the authors bridge a critical gap between physics-informed modeling and data-driven ROMs. The training and evaluation procedures are well-defined, and experimental results appear thorough and convincing. The paper is generally well-written and logically organized. The abstract and introduction clearly articulate the motivation and novelty. Figures and results are presented in a digestible manner.\n\nWeakness: I have two major concerns. 1. While Φ-ROM introduces a physics-informed mechanism by incorporating differentiable solvers into the latent dynamics, it’s unclear whether this approach offers a fundamentally better alternative to PINN-ROM. PINN-ROM directly enforces the true PDEs during training—even if it performs worse empirically—its objective remains aligned with the exact continuous PDE. In contrast, Φ-ROM relies on a discretized solver whose numerical errors (local and global truncation errors, as understood from Taylor expansion analysis) mean the model is optimizing toward a discretization-dependent surrogate. This raises the concern that Φ-ROM may inherit the solver’s biases rather than learn the true physics. The paper does not sufficiently justify why this tradeoff (solver-consistency over PDE-consistency) should be preferred in principle. 2. The Φ-ROM architecture seems to fall into a regime that the original CROM paper explicitly discouraged—namely, building a low-dimensional representation (\\alpha) from already discretized vector fields. CROM advocated modeling the continuous dynamics of the field itself rather than that of a discretized approximation. Yet, this paper claims Φ-ROM outperforms CROM broadly, which introduces a tension: either Φ-ROM violates the key assumptions behind CROM, or CROM’s earlier recommendations do not hold universally. The authors do not discuss this contradiction, nor clarify what might enable Φ-ROM to work better in such a regime. A clearer reconciliation between these two frameworks is warranted.\n\nQUESTIONS:\nTwo questions:\n- The authors mentioned \"When trained, the latent coordinates corresponding to a new state are computed by inversion\", \"The\nminimization problem in Eq. (4) is solved iteratively using a gradient-based optimizer.\" -- why didn't the author build an encoder to construct the \\alpha from u, which is probably more common in auto-encoder based ROM literatures?\n- For the description of Fig 1: \"The initial condition and the forecast solution at a target time T∗, can be observed and reconstructed on arbitrary (and possibly different) grids X and X′\" -- why the two grids could ever be different?\n\nLIMITATIONS:\nYes. The author explicitly mentioned \"we acknowledge that requiring access to differentiable PDE solvers limits Φ-ROM’s immediate\napplicability\" which is absolutely true. And I don't think there's any major limitations for the method beyond that.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-19"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a new network-based method for reconstructing and predicting dynamics governed by spatial-temporal PDEs. It combines two network models: 1) a network that captures the solution manifold with reduced spatial dimensions; 2) a network that learns the temporal evolution of the reduced coordinates on the solution manifold. The training loss of the second network requires a differentiable PDE solver that provides temporal derivatives as supervision. The paper evaluates the proposed method on five 2D PDE problems and compares it with several standard baselines.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. This paper’s writing is pretty good. In particular, I like the background and method sections, which achieve a good balance between telling the high-level story and explaining the low-level technical details.\n2. The design of the network method is nice and clean. There are a number of ways of incorporating neural networks into reduced-order modeling, and I feel the overall design of the two network models provides an intuitive way of capturing the reduced-order dynamics. It is also a fairly general framework that can be applied to various PDEs and discretization schemes.\n\nWeaknesses:\n1. The experiments only consider two-dimensional PDE examples with low-resolution discretization schemes, i.e., roughly a hundred by a hundred cells according to the appendix.\n2. The experiments seem to lack comparisons with numerical PDE solvers. The motivation and benefits of applying neural physics over using existing numerical solvers are unclear to me.\n\nQUESTIONS:\nRegarding the technical method:\n1. The method’s dependence on a differentiable solver seems to arise from the dynamic loss. As Remark 3.2 suggests, one could replace this loss design with the true gradients computed from the underlying PDEs, like in PINNs. To me, this seems to be a better alternative as it avoids the dependency on a differentiable PDE solver and exploits the true time derivatives from the underlying dynamics. So, have you considered this option as suggested in Remark 3.2? Would this alternative achieve comparable or better performance than your proposed method?\n\n2. The hyperparameter lambda is a bit unnatural and mysterious to me. Could you provide an ablation study on the effects of different lambda values and some insights for choosing them in a new problem?\n\n3. The reconstruction and dynamic losses characterize the spatial and temporal errors, respectively. I wonder whether they could be viewed through the lens of some variational or weak forms of the underlying PDEs, which may shed more light on the physical meaning of lambda.\n\nRegarding the experiments:\n\n4. Are there any technical reasons that may prevent the proposed method from being applied to 3D, larger-scale PDE problems?\n\nLIMITATIONS:\nI appreciate that the limitation section is up front about its dependency on a differentiable solver.\n\nI would suggest the limitation section stress that the current experiments are limited to 2D problems with small grid resolutions.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a Physics-informed Reduced-Order Model that builds on the DIno framework and directly trains on latent dynamics using the labels from the differentiable PDE solver. Numerical results show the advantages of this method compared to DIno and other physics-informed ROMs.\n\nSTRENGTHS AND WEAKNESSES:\n[Strengths]\n\nS1) Numerical experiments are extensive, including five representative problems and five baselines. \n\nS2) Comprehensive appendix with computation details, dataset description, hyperparameter setups, and visualizations.\n\nS3) The proposed method generalizes much better compared to DINo.\n\n[Weaknesses] \n\nW1) The computation of $\\dot{\\alpha}^{\\star}$ brings restrictions to the methods (requiring differential PDE solvers) and additional computation cost. The tradeoff between accuracy and computational cost for different baselines considered in this paper has not been fully discussed and compared.\n\nW2) There is no guidance on how to choose the reduced latent dimension $k$. The error brought by hyper-reduction for solving $\\dot{\\alpha}^{\\star}$ could be hard to quantify, resulting in learning inaccurate latent dynamics.\n\nQUESTIONS:\nQ1) How well does the method perform compared to non-ROM methods? Including some non-ROM baselines (like Physics-informed neural operator [1] and Physics-informed deep operator network [2]) may help readers get a more comprehensive sense of the position of the proposed method in the broader field of physics-informed machine learning. \n\n[1] Li, Zongyi, et al. \"Physics-Informed Neural Operator for Learning Partial Differential Equations\"\n\n[2] Wang, Sifan, et al. \"Learning the solution operator of parametric partial differential equations with physics-informed DeepONets.\"\n\nQ2) Why we observe smaller forecasting errors in $[T_{tr}, T_{te}]$ compared to in $[0, T_{tr}]$ in Diffusion case of Table 1? Typically, model predictions on the training dataset should have a smaller prediction error compared to the test dataset. \n\nQ3) Could you further explain the reasoning for the choice of $\\lambda$, which depends on the sensitivity of the differentiable solver to the noise and errors in the reconstructed fields?\n\nLIMITATIONS:\nYes, adequately described in the paper.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes \\Phi-ROM, a physics-informed reduced-order modeling (ROM) framework that leverages differentiable numerical solvers to guide the learning of latent space dynamics for time-dependent and parameterized PDEs. Unlike prior ROM approaches that treat numerical solvers as mere data generators, \\Phi-ROM integrates them into the training loop, ensuring that the learned latent dynamics remain consistent with the discretized governing physics. \n\nThe model architecture builds on an existing framework, using an implicit neural representation (INR) decoder and a dynamics network to evolve latent representations over time. By using automatic differentiation (AD) and hyper-reduction, \\Phi-ROM achieves computational efficiency while being mesh-agnostic, allowing training and inference on different spatial discretizations. The authors demonstrate \\Phi-ROM's performance on a variety of 1D and 2D PDEs using diverse solvers (finite difference, finite volume, and LBM), and show improved generalization to unseen parameters and sparse observations compared to baseline ROM and PINN-style models.\n\nSTRENGTHS AND WEAKNESSES:\n*Strength*\n\n- The paper contributes a new category of ROMs that use differentiable solvers, moving beyond traditional projection-based or PINN-augmented methods. The use of differentiable solvers as part of the training process addresses a key limitation of purely data-driven ROM (their divergence from physical constraints during long-time evolution).\n\n- The coupling of hyper-reduction with latent dynamics training is non-trivial and is a potentially novel contribution in the ROM literature.\n\n- The empirical comparisons are extensive and systematic. The authors test their model across various PDE types, solvers, and settings (dense vs. sparse, interpolation vs. extrapolation).\n\n- The mesh-free (or rather the mesh-agnostic) property of the method is particularly appealing for scientific computing pipelines where spatial discretizations evolve over time (e.g., AMR, or moving meshes).\n\n*Weakness*\n- The set of benchmark problems is limited to relatively simple PDEs with constant coefficients and low spatial dimensionality (1D and 2D). These are useful for validating the method but fall short of demonstrating its utility in complex, real-world scenarios, which is arguably the key motivation for such physics-informed ROMs.\n\n- While the components (INRs, differentiable solvers, latent dynamics) are not new individually, their composition in this context is original. Still, more emphasis could be placed on how this method fundamentally differs from DINo or PINN-ROM beyond training strategy.\n\nQUESTIONS:\n- All test cases involve constant-coefficient PDEs. Can the authors demonstrate performance on PDEs with spatially varying diffusivity, anisotropic tensors, or stiff source terms?\n\n- What is the bottleneck in extending this approach to 3D or higher-resolution simulations? Can the authors comment (or been better, demonstrate) an example in 3D (see next point)\n\n- Given the reliance on Jacobians and inversion (even with hyper-reduction), does \\Phi-ROM scale to high-dimensional (e.g., N ~ 10⁶) settings (of mesh size)? What are the memory/runtime implications?\n\n- Can the method support adaptive spatiotemporal sampling during inference? Sampling more near high gradients, for instance?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work proposes a physics-informed reduced order modeling to simulate fundamental PDEs, including Diffusion, Burgers, Navier-Stokes, KdV, and LBM equations. The approach integrates differentiable solvers into the training process to constrain the latent dynamics and ensure consistency with governing physics. The paper validates the efficacy of the proposed framework for interpolation within the training time horizon and extrapolation beyond training time on unseen initial conditions and parameters. The method is also evaluated under sparse observation settings to demonstrate its robustness.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. The method presentation is clear, and the training procedure and architecture description are well structured.\n\n2. The authors provided the code, which supports reproducibility and further experimentation.\n\n3. Several PDE test cases with varying levels of complexity are considered, and several comparisons with existing approaches are provided.\n\nWeaknesses:\n\n1. Comparisons with methods focused on extrapolation tasks are missing. Several neural PDE solvers already address the challenge of extrapolation, such as PDE Refiner. A comparison or discussion with respect to those methods is missing in the paper.\n\n2. Training time and computational cost of the proposed method are not discussed or benchmarked, which is essential for assessing practical feasibility.\n\nQUESTIONS:\n1. This work is dependent on how latent dynamics evolve over time. How could this framework be adapted or modified for the PDEs not reliant on time? Such as elliptic PDEs\n\n2. Operator learning methods have recently shown success for extrapolation tasks. There are no comparisons with recent operator learning methods, such as convolutional neural operators. Also, a comparison with the method PDERefiner would enable to assess the potential of the proposed method in an extrapolation scenario. Could the authors comment on this and explain how Φ-ROM compares to these methods?\n\n3. The discussion mainly focuses on accuracy for interpolation and extrapolation tasks. Since this framework evaluates trajectories over time, how does training time scale with the increasing complexity of PDEs, higher spatial dimensions, or higher-order systems?\n\n4. The training time horizon $T_tr$ and testing time horizon $T_{ex}$ differ for each test case. What are the criteria for selecting these time intervals, and how sensitive are the results to this choice?\n\nLIMITATIONS:\nYes, discussed", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}