Title: Learning mechanical systems from real-world data using discrete forced Lagrangian dynamics
Venue: NeurIPS 2025
Accepted: no
Keywords: physics informed

structure preserving

lagrangian

hamiltonian

symplectic

energy conservation

non-conservative systems

dissipative dynamics

variational integrator
Abstract: We introduce a data-driven method for learning the equations of motion of mechanical systems directly from position measurements, without requiring access to velocity data. This is particularly relevant in system identification tasks where only positional information is available, such as motion capture, pixel data or low-resolution tracking. Our approach leverages the discrete Lagrange-d’Alembert principle and the forced discrete Euler-Lagrange equations to construct a physically grounded model of the system’s dynamics. We decompose the dynamics into conservative and non-conservative components, which are learned separately using feed-forward neural networks. In the absence of external forces, our method reduces to a variational discretization of the action principle naturally preserving the symplectic structure of the underlying Hamiltonian system. We validate our approach on a variety of synthetic and real-world datasets, demonstrating its effectiveness compared to baseline methods. In particular, we apply our model to (1) measured human motion data and (2)  latent embeddings obtained via an autoencoder trained on image sequences. We demonstrate that we can faithfully reconstruct and separate both the conservative and forced dynamics, yielding interpretable and physically consistent predictions.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a data-driven approach to learn the equation of motions of dynamical systems. The approach addresses the following three challenges:\n- Preservation of the variational structure of the dynamics, that is being generated from a principle of least action.\n- Trajectory data consisting only of positions, but not velocities or momenta.\n- The presence of dissipative forces.\nThe Authors combine ideas from previous works on Lagrangian Neural Networks, ad well as variational integrators to define a (regularized) loss function to minimize at training time, as well as an autoregressive inference pipelines. \n\nThe approach is tested on four datasets of moderate dimension, showing reasonable performance.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n- The approach is principled and easy to understand. Thanks to automatic differentiation, I guess it is also easy to implement. \n- The paper is clear and well-written, with an adequate literature review.\n#### Weaknesses\n- In my judgement, the paper is a small improvement over the state of the art, essentially packaging ideas from [a, b, c] into a single algorithm. The main contributions, which are stated in Section 1.2, include (a) which is not novel (see Ref [a]), (b) and (c) that are arguably _not_ contributions but features of the model, (d) which is the experimental evaluation and (e) which I think is an original additional experiment but frankly not aligned with the main message of the paper.\n- I also have some concerns on the experimental evaluation, and specifically with the lack of the baseline from ref [a]. I actually think that the baselines presented are well chosen:\n\t- Neural ODE is a general-purpose dynamical model\n\t- GLNN includes dissipation, but not the variational integrators\n\tThat's why I would have expected the presence of LSI from [a], since in some respects it is complementary to GLNN: includes variational integrators but not dissipation.\n\nGiven these comments, I do not recommend the publication at NeurIPS. Despite the paper being well-written and principled, I do not find enough novelty to justify a recommendation of acceptance.\n\n[a] Sina Ober-Blöbaum and Christian Offen. *Variational learning of euler–lagrange dynamics from data.*\n\n[b] Miles Cranmer, Sam Greydanus, Stephan Hoyer, Peter Battaglia, David Spergel, and Shirley Ho. *Lagrangian neural networks.*\n\n[c] J. E. Marsden and M. West *Discrete mechanics and variational integrators.*\n\nQUESTIONS:\n- Can you clarify more concretely what you mean by \"structure-preserving\" models? I interpreted it as a model that can be linked to a principle of least action, but I don't know if I understood correctly.\n- You write: \"In practice, velocity and momentum estimates are usually approximated from sequential position measurements using finite differences, which leads to inaccuracies due to noise and truncation errors\". Isn't this true for this method as well? At the end of the day, you define $\\bar{\\dot{q}}\\_{n + \\frac{1}{2}} = h^{-1}(q\\_{n + 1} - q\\_{n})$\n- In generating the datasets, you wrote: \"Training data for these tasks are generated by solving the corresponding analytical ODEs with Gaussian noise applied to each sample.\" Why the Gaussian noise?\n\nLIMITATIONS:\nSome limitations are discussed in the supplementary material, but they only address the experimental evaluation and not the whole approach.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-17"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the problem of learning the dynamics of physical systems directly from position data. It proposes using neural networks to represent the Lagrangian and Lagrangian force functions within the formalism of Lagrangian mechanics. While this is a well-studied problem, previous methods require velocity data, which is often obtained through numerical approximations. Instead, the authors formulate a loss function using a discrete version of the Lagrange–d'Alembert principle. Minimizing this loss generates physically consistent trajectories. The method can also be applied to high-dimensional measurements, such as images, by first mapping them to a latent space using an autoencoder. The authors validate the method on three synthetic data experiments and one real-world data experiment. The experiments suggest that the proposed method outperforms the alternatives considered.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n\n- To the best of my knowledge, the submission is technically sound, and the proposed methodology is reasonable.\n- The paper provides supporting evidence for its two main claims from both real-world and synthetic data experiments: that the method is more performant for predicted rollouts, and that the model can separate conservative and dissipative dynamics.\n- The paper is clear and generally well-written; the figures effectively communicate the model's structure.\n- The mathematical exposition is clear and appropriate for a general audience.\n- The proposed method appears to be an advancement on previous work and allows these models to be applied to more general datasets (i.e., from purely positional measurements).\n\n## Weaknesses\n- The model is only compared to the GLNN model. It seems that other models could also be suitable for this data, for example, [Dissipative SymODEN](https://arxiv.org/abs/2002.08860) or the Deep Dissipative model proposed in [\"Learning Deep Dissipative Dynamics\"](https://arxiv.org/abs/2408.11479). Both of these works use similar benchmark problems, so they are comparable to the proposed approach. It is difficult to evaluate the impact of the proposed approach without these comparisons.\n- The experiments are somewhat limited and consist mostly of simple synthetic systems.\n- The contribution, although significant, appears incremental when compared to existing approaches.\n\nQUESTIONS:\n- The discretized Lagrangian system uses a midpoint approximation to form the generalized velocity coordinates. How is this approximation fundamentally different from using the numerical derivative approximations found in prior work?\n- This is a minor point, but the plots, particularly Figure 3, are too small and require significant zooming to be legible.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-24"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a data-driven framework to learn the equations of motion of mechanical systems based on discrete d’Alembert principle and forced Euler-Lagrange equations. With proper formulation, the proposed system brings the following contributions: (1) it can learn from position data only without the need of velocity or momentum data; (2) it preserves the system structure and is intuitive to incorporate dissipation terms such as external forces.\n\nSTRENGTHS AND WEAKNESSES:\n### **Strengths**\n\nI am not an expert in dynamical systems, so I may have missed some mathematical details and did not verify all derivations in detail. Based on my understanding, the main strength of this paper lies in the following:\n\n[A novel theoretical formulation with improved experimental results]\n- I feel this work is more like a theoretical paper which presents a theoretically grounded discrete formulation for learning dynamical systems, distinguishing it from many prior methods. While I may not fully understand all aspects of the formulation, the experimental results suggest that the proposed framework is both more stable and more accurate than existing methods, such as Neural ODEs and GLNN.\n\n### **Weaknesses**\n\n[Visualization]\n- I’m more familiar with the motion capture stuff. Based on my understanding, the proposed framework can use an autoencoder architecture to learn the underlying dynamic system in the latent space, thereby enabling motion reconstruction and rollout for future motion prediction. To demonstrate this claim, I would say at least a supplemental video should be necessary, including both the ground truth and reconstructed motion sequences and future predictions. The current visualization in Figure 5 is a bit hard to tell the differences, especially between the proposed method (the second row) and Neural ODE (the last row).\n- Similarly for the pendulum task, a video demonstration of all the compared methods should be helpful to further show the difference between them.\n\nAlso I feel the format of this submission could be improved, which I will list below.\n\nQUESTIONS:\nI do not have specific technical questions regarding this submission. Given my limited expertise in this area, I will refrain from being overly harsh and currently lean toward acceptance. I look forward to the rebuttal phase to hear perspectives from other reviewers and hope the authors will address the weaknesses and formatting issues I listed.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a Lagrangian Neural Network (LNN) formulated in discrete time that is also able to learn non-conservative forces, such as dissipation/damping or external forces, from data. The approach is, for example, validated both in state space for learning the motion of a double pendulum, a charged particle in a magnetic field, and human motion capture data, as well as in learned latent space for learning the motion of a pendulum based on image observations.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n\n- Reducing the need for velocity data when training LLNs indeed seems like a worthwhile goal, as velocity data is often not available in real-world datasets/systems (without finite differences or integration of IMU data that tends to exhibit drift).\n- The method is extensively validated on a variety of tasks, including state space and latent space learning, and both quantitative and qualitative results are strong.\n\n## Weaknesses\n\n- The specific difference of the proposed method (both in terms of rollout/inference, and training loss) with respect to existing continuous-time LLNs trained with Euler-forward discretization is not obvious and should be clarified via a side-by-side comparison.\n- Some of the derivation steps are not explained in detail, which makes it hard to follow the derivation of the equations. See the questions section for more details.\n- The performance of the baseline methods, specifically the existing LNN/GLNN, seems surprisingly bad and unstable and does not match the performance of existing LNNs in the literature. This should be clarified further. Is it an implementation/hyperparameter issue?\n\n### Detailed Comments on Minor Issues\n\n- Figure 1 could be slightly enlarged to make the text more readable.\n- Table 1: Add definitions of the abbreviations of the various methods in the table caption.\n- Many of the contributions statements/bulletpoints sound not like actual contributions, but rather like descriptions/features of the method (e.g., b, c). Contributions should claim something novel with respect to the existing literature, not just describe the method.\n- For reference [42], the authors should cite the ICLR or IJRR versions of the paper, not the arXiv preprint.\n- Table 2: Please report over how many random seeds the statistics (mean and standard deviation) are computed. Also, please directly cite the baseline methods in the table.\n- Line 187: The term \"Extrapolation error\" seems a bit misleading, as it implies generalization to data unseen in the training distribution. Instead, this seems more like a standard \"Rollout error\"/\"Trajectory prediction error\".\n- Figure 3: Why is panel (a) labeled with \"damped\" and panel (c) labeled with dissipative? What is the difference between these two terms in this context? Please clarify.\n- Lines 266-267: The claim that the method learned \"generalizable governing equations\" seems questionable as the method was only trained on two trials of motion data of a single subject.\n\nQUESTIONS:\n- One of my largest doubts about this paper is the difference between the proposed method (that claims that it does not require access to velocity data or finite differences) and vanilla Euler-forward discretizations of existing continuous-time LLNs. Clarifying these aspects would be crucial for my final recommendation on this paper. For example, even though the authors claim that that finite differences for estimating the velocity are not needed, they exactly do that, for example in Equations (7) and (8) via the term $\\frac{q_{n+1}-q_n}{h}$. Also, can the authors please prepare a side-by-side comparison of the proposed method with a vanilla Euler-forward discretization of an existing continuous-time LNN (e.g., Lagrangian Neural Networks, or LNNs)? What exactly are the differences during training and rollout?\n- Can you please explain in more detail the derivation of Equation (5) and extend the explanation in the manuscript (i.e., show more steps of the derivation)?\n- Section 3.3, lines 164-165: Can the method also simultaneously learn external forces and dissipation? If so, how would this be done? If not, why not?\n- Section 4.1, line 211: How does the magnitude of the measurement noise compare to the magnitude of the states $q$. Could you show a plot to visualize this?\n- My most significant doubt about the results (Section 4.1): It seems very surprising to me that GLNN is so unstable, while all the other methods do not exhibit this issue. Why is this the case? Is this also the case when you remove the measurement noise? Is the GLNN trained without trajectory rollouts? Is the NeuralODE trained with trajectory rollouts? To me, it seems very unintuitive that the GLNN is unstable, while the NeuralODE is stable. Could you please clarify this point further?\n- Line 217: Why do the baseline methods struggle to generalize to the conservative regime? Is it because you applied measurement noise to the training data?\n- The authors specify that the method was trained in Task 4 on a two trials of human motion capture data. Why did you use not train on more trials, on multiple subjects, or even on the full dataset?\n\nLIMITATIONS:\nThe authors claim in the NeurIPS paper checklist that the limitations are discussed in the appendix. However, the appendix is not included in the PDF. Please add the appendix to the PDF. In any case, please also summarize the limitations in the conclusion section of the main text.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-03"}