MetaSym: A Symplectic Meta-learning Framework for Physical Intelligence
Authors: Pranav Vaidhyanathan
Aristotelis Papatheodorou
Mark T. Mitchison
Natalia Ares
Ioannis Havoutis
Venue: Accepted by TMLR ()

Abstract: Scalable and generalizable physics-aware deep learning has long been considered a significant challenge with various applications across diverse domains ranging from robotics to molecular dynamics. Central to almost all physical systems are symplectic forms, the geometric backbone that underpins fundamental invariants like energy and momentum. In this work, we introduce a novel deep learning framework, MetaSym. In particular, MetaSym combines a strong symplectic inductive bias obtained from a symplectic encoder, and an autoregressive decoder with meta-attention. This principled design ensures that core physical invariants remain intact, while allowing flexible, data efficient adaptation to system heterogeneities. We benchmark MetaSym with highly varied and realistic datasets, such as a high-dimensional spring-mesh system  Otness et al. (2021), an open quantum system with dissipation and measurement backaction, and robotics-inspired quadrotor dynamics. Crucially, we fine-tune and deploy MetaSym on real-world quadrotor data, demonstrating robustness to sensor noise and real-world uncertainty. Across all tasks, MetaSym achieves superior few-shot adaptation and outperforms larger state-of-the-art (SOTA) models.

Published in Transactions on Machine Learning Research (02/2026)
MetaSym: A Symplectic Meta-learning Framework for Physical Intelligence
Pranav Vaidhyanathan" Deportment of Engineering Science University of Ouford, UK
pranav@robotsooz.cc..
Aristotelis Papathcodorou" Department Engineering Science University of Oxford, UK
aristotelle@rrooossoooccc
Mark T. Mitchison
mar..mitchismmnkckc.ccco
School of Physics, Trinity College Dublin, Ireland Department Phayics, King's s College London, UK
Natalia Ares
Department Engineering Science Universily of Oxford, UK
Toannis Havoutis
kounmis@robots.orz
Department of Engineering Science University of Oxford, UK
Reviewed OpenReview: https:// openrreveee. net/ forum? id=M/1uf1
Abstract
Scalable and generalizable physics-aware deep learning has long been considered a signif- icant challenge with various applications across diverse domains ranging from robotics to molecular dynamics. Central to almost all physical systems are symplectic forms, the geo- metric backbone that underpins fundamental invariants like energy and momentum. In this work, we introduce novel doep-learning framework, MetaSym. Our approach combines strong symplectic inductive bias obtained from symplectic encoder, and an auloregressive decoder with meta-attention. This principled design ensures that core physical invariants re- main intact, while allowing flcxible, data efficient adaptation to system heterogencitics. We benchmark MetaSym with highly varied and realistic datasets, such as a high-dimensional spring-mesh system (Otness et al., 2021), an quantum system with dissipation and mea- surement backaction, and robotics-inspired quadrotor dynamics. Crucially, we fine-tune and deploy MetaSym real-world quadrotor data, demonstrating robustness to sensor noise and real-world uncertainty. Across all tasks, MetaSym achieves superior few-shot adaptation and outperforms larger statc-of-the-art (SOTA) models.
1 Introduction
Learning to predict the dynamics of physical systems is fundamental challenge in scientific machine learn- ing, with applications ranging from robotics, control, climate science, and quantum computing (Ghadami & Epureanu, 2022; Zhang et al., 2024: Alexeev et al., 2024). Traditional approaches often rely on care- fully derived differential equations (Papatheodorou et al., 2024) that embed known conservation laws and gcometric propertics (c.g., Hamiltonian or Lagrangian mechanics). Although these classical models have
"Equal Contribution

---

Published in Transactions on Machine Learning Research (02/2026)
been tremendously successful in capturing fundamental physics, they can become unwieldy or intractable when faced with complex real-world phenomena, such as high-dimensional systems, intricate interactions, or partially known forces, that defy simple closed-form representations. Recent progress in deep learning has opened new avenucs for data-driven modeling of dynamical systems, bypassing the need for complete analytical descriptions (Carleo el al., 2019). Neural ODE frameworks such as Chen el al. (2019), for instance, reinterpret dynamic evolution as a continuous function learned by a neural network, while operator-learning approaches such as Fourier Neural Operators (FNOs) in Li et al. (2021) allow for flexible mappings from ini- tial conditions to solutions of partial differential equations. Despite these advances, deep learning approaches often face two critical challenges:
Preserving the underlying physical structure. Standard networks, left unconstrained, may inadvertently violate symplectic forms, conservation of energy, or other geometric constraints intrin- sie physical dynamics (Chen al., 2020). These violations can accumulate over time, producing qualitatively incorrect long-horizon predictions, c.g. spurious energy drift in Hamiltonian systems.
Generalizing across related systems. Many real-world applications involve entire families of similar yet distinct systems (e.g., variations of robotic manipulator differing in load mass or joint friction, or molecular systems differing in exact bonding parameters). Training an entirely separate model for each variant is both data-inefficient and computationally expensive. Without mechanisms to share knowledge, a network trained on one system will typically fail to adapt efficiently to another, even if they exhibit similar physical behaviors. Bridging this sim-2-real gap is crucial for variety of tasks such as control and real-time prediction Bai al. (2024).
The trade-off between flexibility (i.e., capacity to Iearn diverse dynamics) and the enforcement of physical constraints can be addressed through specialized architectures that embed geometric priors (Chen et al., 2020; Greydanus et al., 2019) related to the physical system. Notably in the context of Hamiltonian mechanics, the Symplectic Networks (SympNets), introduced in Jin et al. (2020), incorporate symplectic forms directly into their design, guaranteeing that learned transformations represent canonical transformations. This preserves key invariants such as energy and momentum, mitigating error accumulation in long-horizon forecasts.
However, woaldworld systems also exhibit heterogeneity, i.e., varying parameters, boundary conditions, even control signals that deviate from conservative dynamics. Thus, meto-learning becomes natural extension (Schorling et al., 2025). By training on a set of related systems, meta-learning-bassd methods (e.g., Model- Agnostic Meta-Learning (MAML), interpretable Meta neural Ordinary Differential Equation (iMODE) and Fast Context Adaptation Via Meta-Learning (CAVTA) (Finn et al., 2017; Li et al., 2023; Zintgral et al., 2019b)) acquire high-level inductive biases that can be quickly adapted to novel systems using limited additional data. Consequently, when faces new variant of familiar system, the network can fine-tune handful of parameters, rather than retrain from scratch. This provides robust and scalable performance.
1.1 Contributions
In this work, we introduce MetaSym, a deep-learning framework that addresses the major challenges of data-driven modeling of physical systems, ie., preserving underlying geometric structures to ensure physically consistent behavior over long time-horizon and rapidly adapting to system variations with minimal data. Our contributions are listed below:
Symplectic encoder for structure preservation. We propose a specialized neural encoder (SymplectieEncoder)) built on SympNet modules. The inherent structural invariants of the SympNets provide strong inductive bias to the SymplectidFncodder while our bi-directional training pipeline enforces Hamiltonian consistency, obtained by the canonical transformations that pertain to different systems. Hence, the SymplecticEncodo's output conserves key geometric invariants (c.g., enorgy and mm effectively minimizing error accumulation over long-term roll-outs with minimal architecture size.
Autoregressive decoder with meta-attention for adaptation. To handle non-conservative forces and variations in system parameters, introduce ActiveDecoder, transformer-inspired de-
2

---

Published in Transactions on Machine Learning Research (02/2026)
coder module equipped with meta-attention framework. This decoder incorporates control inputs, external forces, and per-system parameters enabling flexible modeling of real-world effects beyond ideal Hamiltonian dynamics, while enabling autoregressive multi-step prediction during inference time.
Meta-learning for multi-system generalization We adopt different meta-learning paradigms for the SynalecticEnccder and AcliveDecoder of our architecture motivated by specific design choices explained in Section 3. This yields a single framework that quickly adapts new or modified systems, even in real-world scenarios.
By integrating physical constraints and generalizable architectural changes into novel training pipeline that utilizes meta-learning updates for both our SymplecticEncoder and DetiveDe we benchmark this bespoke smaller architecture against other SOTA deep learning methods, including Dissipative Hamiltonian Neural Networks (DHNNs) (Sosanya & Greydanus, 2022) and Transformers (Vaswani, 2017; Geneva & Zabaras, 2022), for modcling various physical systems in both classical and quantum regimes. These systems include high-dinenniooal spring-mesh system, an open-quantum system whose dynamics are highly complex and counterintuitive in the classical regime, and a quadrotor with floating- base dynamics. This provides evidence of far reaching implications for diverse set of physics modeling tasks including challenging tasks like real-time quantum-dynaninn prodiction and simulating a variety of complex dynamics that typically require complex computalional methods to solve. To the best of our knowledge, MetaSym is the first bespoke physics-basee deep-learning model to adapt and generalize well to both classical and non-unin non-unitary quantum dynamics.
2 Related Work and Background
Physicists have long utilized the Lagrangian and Hamiltonian formalisms of mechanics to study the dynamics of physical systems (Kibble & Berkshire, 2004). Consider q (q....., qa) that represents the generalized coordinates of an d-dimensional configuration space (position), while p represents the corresponding gener- alized momenta. We can describe the Hamiltonian H(q, p, () as the total energy of the system. This leads to Hamilton's 's equations as follows:
OH 9. 0p:"
0H 0q.
i= i=.....,d d.
(1)
This formalism naturally imbues the phase space (q, p) E 2d with geometric structures that are vital to the study of these physical systems. One such structure is the symplectic form (w) given by, w = - Ap;A dg:- Concretely, a map o, : (q(0), p(0)) + (q()),p(() is said to be symplectic if diw = w. This implies that the flow in the phase space preserves the volume (Royer, 1991).
Inspired by such geometric formulations of mechanics, recent work such Hamiltonian Neural Networks (HNNs) and Lagrangian Neural Networks (LNNs) have sought to embed physical priors into deep-learning architectures (Greydanus et al., 2019; Cranmer et al., 2020). In particular, SympNets have emerged as structure-preserving neural architectures designed specifically for learning Hamiltonian systems, ensuring that their learned maps are intrinsically symplectic. These architectures admit universal approximation the- orems and crucially, this construction does not require solving ODEs or differentiating a candidate Hamilto- nian during training and inference times, which often leads more efficient optimization compared to other architectures such as HNNs or LNNs. The collection of all SympNets forms group under composition, ensuring that every learned map is invertible with closed-form inverse. However, due to the fundamental nature of Hamiltonian systems and symplectic forms, SympNets and similar architectures fail to generalize to dissipative systems where the symplectic structure is no longer preserved (Chen et. al., 2020; Cranmer et al., 2020; Jin et al., 2020). While there have been attempts to reconcile dissipative dynamics and control inputs to model realistic physical systems by preserving the symplectic framework, such as the Dissipative SymODEN (Zhong et al., 2020; 2024), they often suffer from the lack of generalization to different systems (Okamoto & Kojima, 2024).
3

---

Published in Transactions on Machine Learning Research (02/2026)
Generalization between different but related physical systems is also vital for deep-learning methods to excel at physics modeling. Meta-learning serves as the natural avenue for exploring such strategies. Building on series of meta-learning strategies (Finn et al., 2017; Rajeswaran et al., 2019; Zintgraf et al., 2019a; Nichol, 2018), the iMODE framework represents a notable advancement applying deep learning to families of dynamical systems. In this selup, one set. of parameters captures the universal dynamics shared across all systems in family, while another set encodes the idiosyncratic physical parameters that differentiate one system instance from another. However, certain drawbacks still persist. Apart from the lack of physics priors, the existing meta-learning approaches, such iMODE, suffer from lack of scalability (Choe et al., 2023).
Frozen during inference
Teacher forcing
Spring Meah
Quadrotor
Quantum Sysstmm
Attention Parameters
MLP Parameters
Physical Systems
SympNett
0,0000
Norm
Norm
Norm
Values
Attention
OrgePen Ae)
G0
Cross Attention
MLP
SympNet
Stop-grad
SymplecticEncoder
ControlNet
ActiveDecoder
Autoregressive.com step
Figure 1: MetaSym integrates a Sympleclic (light-green), AcliveDecoder (light-purple), and Con- trolNet (pink). The SymplecticEncoder is pre-trained in isolation on conservative state-space data using shared newward/inveree networks receiving (q. (t) (i) At) and (q
P++1. (5) P At) respectively, and thus explic-
PP:
itly enforcing time-reversibility. Subsequently, with the SymplecticEncodee frozen, the ActiveDecoder and ControlNet are then trained autoregressively via teacher forcing, where system-specific adaptation is achieved by fine-tuning the cross-attention's query/valut parameters (purple dots in cross attention) with few-shot gradient steps. During inference, the ControlNet processes a sequence of non-conservative coordinates and control signals (8) Pe:T: At, (,T), while the SymplecticEncoder projects them onto the conservative man- ifold and integrates them on it, producing z.- The AcliveDecoder using its cross-attention, perturbs 2, to predict the dynamics the system for future time-steps, autoregressively. We indicate the next-step predic- tions of our network with (9.411 (1:11.0(11)
3 Methods
In this section, we detail our pipeline for learning, adapting, and predicting the dynamics of physical systems using MctaSym, our structure-preserviig neural architecture and meta-lcarning framework. We describe both the high-level design of encoder decoder model and the specialized training procedures implemented via meto-learning. Fig. represents the overview of the architecture. Details on the described training and meta- learning pipelines are given in Appendix A, while each design choice is based ablation studies provided in Appendix D.
3.1 SymplecticEncoder: Structure Preservation
To ensure that predictions preserve fundamental geometric invariants, the encoder module is implemenned as SympleeticEncooerr Internally, it uses symplectic neural network (e.g., LASympNet by Jin et al. (2020))
consisting of sub-layers that update position ( and momentum P designed to approximate Hamiltonian flows and preserve symplectieity. Specifically, each sub-layer either
(5) for systems 1,..., in manner

---

Published in Transactions on Machine Learning Research (02/2026)
performs an "up" or "low" transformation,
(Up) dr + q: ap,
(8)
(1)
(9) At,
(6) (4)
P + Pi
(2)
(Low) qt q P Pt + 3(g (8) At.
(
(
(0)
Here, we use a and B that can be either linear or activation modules, representing learnable parameters, ensuring we remain in the class of canonical (i.c., symplectic) maps. By stacking multiple up/low blocks, (i) (i) At) (q (1) A crucial mathematical property is that the composition of symplectie transformations remains symplectic. Thus, when these modules are stacked together in sequence, regardless of network depth or configuration, the resulting transformation is guaranteed to be symplectic. This relies on the algebraic properties of the "Up" and "Low" updates, which act as symplectic shear transformations. For instance, the Jacobian matrix Joo of a "Low" update (q.p) + (q.p+ p -()))1) is lower-triangular with identity diagonals:
we achieve deep network ty (q: Pt
1)
Low =
8B 88
At
(3)
For map to be symplectic, its Jacobian must satisfy JJ = S. This condition holds for triangular maps 08) is symmetric matrix. To ensure this, we implement a and 3 either provided the cross-term block (here, oq
gradients of scalar potential or using layers with symmetric weights (Jin et al., 2020).
A pivotal feature of Hamiltonian dynamics is time reversibility, i.e., if we integrate the system forward from (1)
(q p( to (4--1: over time At, then integrating backwards over A should return the system exactly to (q)", Pt
(1) This property lies at the heart of many physical invariants (energy, momentum, etc.)
and is crucial for
stability in forecasting the behavior of such systems.
To replicate this in the neural network, each forward pass provides the update for the forward and inverse network. The forward network is characterized as (q Pt
(I) (1) Ad) = = (9141:Pt-1)) P( and the inverse net-
work @ese acts as not merely the computationally reversed pass of $o but also switches the sign of At (i.c., steps "backwards in time"). From a physical standpoint, prescrving both forward and inverse map enforces the lime-reversal symmetry characterizing Hamiltonian flows. We train single model in both direc- tions simultaneously by sharing gradients between the forward and inverse instances, effectively minimizing roconstruction errors. As a result, the network is less susceptible to artificial energy drift and can better maintain conservation laws over extended forecasts. More specifically, this bi-directional training pipeline acts as implicit regularization by enhancing the model's ability to faithfully approximate the underlying canonical transformations, while allowing consistent backward integration without introducing extraneous numerical artifacts. However, note that only the forward instance, Poges is used during inference.
During training the encoder is provided with sequence of phaso-space points (T time-steps) and encodes them onto a conservative latent space ze = [gene Pene] E 222x that is ultimately used by the ActiveDecoder in the cross attention mechanism to specifically fine-tune the query and value parameters.
Additionally, to ensure that our encoder can generalize quickly across multiple related but distinct physical systems, we adopt MAML-style (Finn et al., 2017) framework. This meta-learning paradigm works par- ticularly well in the SymooeeticEncooder's case, since SympNets represent canonical transformations that are known to have well-defined Hessian (Birtea et al., 2020), which is crucial for methods like MAML that use sccond-order gradients. For each system i in a mini-batch, we split its trajectory into Indapt and Imeta sets. During the fast. adaptation loop, we optimize the parameters of the encoder OSE only on Indapt, by mini- mizing the mean-squared error between its forward predictions and the ground-truth labels. This process simulates "specializing" the encoder to system i's local dynamics using a simple loss function that avoids gradient-terms that may destabilize the subsequent meta-update of Asg. For the outer loop, we perform forward and inverse pass of the Symplectic Encoder and subsequently we minimize combined loss
5

---

Published in Transactions on Machine Learning Research (02/2026)
represented as,
(6) BosE (a: OSE
@
(9) (9+1: P:41: Q
(0)
(0) (6)
Lmota Timeaa Nhatch
[9 q :. p
IELmcta iENbatch
(4)
where meta is the total number of time-steps stored in Imeta set and Natch the total number of meta-learned systems in the batch. While the Crrrplecticennodee trained in isolation, we freeze its parameters during the ActiveDscoder's training.
3.2 ActiveDecoder: Autoregressive Decoder
As mentioned in Section 1.1, we train an autoregrossive decoder to model non-conservative and rcalistic forces such friction, or air resistance, that break pure Hamiltonian symmetries. Unlike the CemplecttiEncotee which strictly enforces canonical updates, the ActiveDecoder can incorporate these extraneous phenomena.
We define 9t and Pt as the dissipative canonical coordinates, of a system An input sequence (qlh) (( At, uese) p(2d+m+1)xe with context-window of с time-steps is provided to linear pro- jection, where u!" Rn represents the m-dimensional control-input driving the system at time This lincar layer, which call ControlNet part of the ActiveDecoder serves to map the input to latent vector za E IR2d representing the non-conservative part of the system's 's dynamics. We then apply masked multi-head self-attention over the sequence for autoregressive decoding. The masking allows us to model the causal dependencies of the decoder's imput sequence.
Next, we apply a cross-attention mechanism, augmented with specialized meta- meta-attention design. To achieve multi-system generalization and fast adaptation, we use a meta-learning setup akin to bi-level optimiza- tion inspired by recent progress (Li al., 2025). We separate the Activelecoder's parameters into global parameters, ADD that remain shared across all systems, and local parameters, Ç that can be interpreted system specific parameters. More specifically, in the meta-attention:
Key parameters remain global and are updated in the outer loop, common to all systems.
Query/Ver Value parameters are re-initialized for each system and fine-tuned with a few iterations the inner loop, allowing the decoder discover per-system or per-task representations.
For each system i, we split the time-sequenced data, q(,, (())) u1() into an adaptation set, Indapt containing Tadapt number of time-steps, and a meta set Imeta with Tmeta number of timo-steps. During the inner loop of the bi-level optimization, we hold OnD fixed and fine-tune G by performing few gradient steps. Each step iterates over Indapt feeding the ActiveDecoder (D) with ground-truth (g Piee) and minimizing an inner MSE loss across Tedapt mumber of time-steps, i.c.,
(1) and control uset
Sinnrr
Σ
(8) |BeAn (qree Pieer ue OAD. C.)
(8) 1q 14+1 P
(
Todapt
This ensures that & adapts to system or task specific idiosyncrasies.
Once the local parameters & have been adapted to the optimal value Çr, we evaluate the model's performance Imeta portion of the trajectory. The corresponding outer loss,
(i) outer
Σ LEImers
DoAD (g
Piter u bei OAD S:) (144), P
(6)
Tmeeta
(() propagates gradients only to OAD. The overall training objective is then the sum over C for all systems i, over the batch. We finalize the decoder with Multi-Layer Perceptron (MLP) that outputs the position and momentum for the next lime-step (q 1it1)
The architecture is then used autoregressively for future

---

Published in Transactions on Machine Learning Research (02/2026)
predictions. The inherent high dimensionality of the ActiveDecoder's internal representations, coupled with the lack of well-defined or tractable Hessian guarantees in this setting (LeCun et. al., 2012), motivates our adoption of this meta-learning paradigm over traditional methods such as MAML. By selectively adapting only a carefully chosen subset of ActiveDecoder's parameters, we are able to minimize both the number of adaplation steps and the quantity of data required during the meta-learning process.
3.3 Autoregressive Inference
Once both the SymplecticEneader and ActiveDecoder (Sections 3.1 and 3.2) are trained, we generate future predictions by rolling out the model auloregressively during inference time. Specifically, at test time, we are given initial phase-space measurements. We first map to the conservative portion of the phase-space, (a ) through the SymplecticEncoder. The ActiveDecoder then produces the next predicted phase-space trajectory (9 P()1), based on the pipeline highlighted in Section 3.2. However, we subsequently treat (9++1 Pit1) as inputs for the next time-steps. This is repeated over the context window provided to the decoder.
In summary, by pairing our SymplecticEncoder (to ensure structure preservation) that provides a strong inductive bias with an Autoregressive Transformer-styl Decoder (ActiveDecoder) equipped with meta- attention (to handle individual system variations), MctaSym can rapidly adapt to diverse physical systems while guaranteeing physically consistent core dynamics. The proof of this strong inductive bias provided by the Enccler and its effect on the training of the ActiveDecoder as part of MetaSym is provided by Appendix C.
4 Results
In this section, we evaluate MetaSym's performance, against a variety of SOTA models and benchmarks, espocially in long-horizon stability and fow-shot generalization. In particular, we report the behavior of MetaSym against Transformers that have achieved impressive performance in modeling physical systems (Geneva & Zabaras, 2022) and DHNNs (Sosanya & Greydanus, 2022) that model systems via the Helmholtz decomposition, although they require gradient information and numerical integrator to obtain trajectory data. Our model predicts the subsequent time-steps directly, which poses a much harder task that does not require numerical integration, which can be time-consuming and may hinder real-time performance. These results also reflect on the effectiveness of our design choices.
As mentioned in Section 1.1, we choose to benchmark MetaSym for three extremely challenging and diverse datasets. Notably, choose the spring-mesh system, which tests scalability and closely resembles finite element modeling of surfaces with different materials. This dataset has provided the SOTA testbed for large dimensional systems (Otness et al., 2021). Moreover, we utilize a well proven, reprodueible dataset of an open- quantum system undergoing heterodyne detection leading to representative quantum effects, such as decoherence, to demonstrate MctaSym's robustness to noise and ficxibility to model all types of physical systems. Finally, we choose to predict the dynamics of a quadrotor, a long-standing benchmark in robotics and challenging due to floating-base dynamics and sensor noise that is simulated using standard Gaussian additive noise (Bansal et al., 2016). Furthermore, we also demonstrate the adaptability of MetaSym real- world data by adapting this model (Mohajerin & Waslander, 2018). The description of each experimental setup is laid in the next sections, however more extensive description can be found in Appendix B. All reported MSE errors are not weighted. In the plots, the errors correspond to each phasc-space coordinate, while in Table 1 the reported errors are calculated across all phase-space dimensions. Finally, information regarding the utilized compute resources and training hyperparameters are provided in Appendices & G.
4.1 Spring-mest mesh System
As our first benchmark, we use the 400-dimensional spring-mest system introduced by Otness et al. (2021), consisting of a 10 x 10 lattice of point masses connected by elastic springs. Each of the 100 masses possesses two positional and two momentum degrees of freedom, yielding 400-dimensional state space. This system,
7

---

Published in Transactions on Machine Learning Research (02/2026)
P. [kg m(a]
x10->
0.5
[m)
Actusl
1.2
G.
0.0
Prediction
3
4
-0.5
1.0
0.5
Nedian laa)
Mestania,)
0.0
0.8
MadiantpJ)
-0.5
Test MSE
Mestanipy)
0.6
p. kkg mkk]
0 (mg)
0.0
-0.5
3
4
5
0.2
Dy [kg
0.0
0.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
8 C
Time [s]
Time [s]
Test Set Systerms
Figure 2: (Left) Time evolution of the system's position and momentum variables for a representative set of masses in the spring mesh. The orange curves represent the ground-truth trajectories for each phase-space coordinate type (i.e., 9y-9y, Px: and Py), while the blue curves depict the model's predictions. The close alignment between these trajectories underscores the model's capacity to accurately capture the underlying long-term dynamics (600 time-steps) of the coupled spring system using a context window of 30 time-steps. (Right) Plots illustrating the mean squared error (MSE) of the time evolution of each phase-space coordinate type (dots) for five (A-E) spring-mess mesh systems in the test set. Each column encapsulates the spread of errors observed across all masses the spring-mesh for given phaso-space coordinate across multiple time-steps, with the boxes marking their median values. The uniformly low median errors across all components demonstrate that the model generalizes effectively to different spring-mesh systems for all phase-space coordinates.
notable for its high dimensionality and complex dynamical couplings, serves as canomical testbed for learn- ing physical systems, including deformable volumes and surfaces relevant to engineering applications (Pfaff et al., 2021). Node positions are updated via forces from spring extensions and compressions, resulting in nontrivial communication and computation patterns across the mesh. For a more in-depth description of the benchmark, refer Appendix B.1. Unless stated otherwise, models are trained using teacher forcing for next-token prodiction with fixed context window of 30 time-steps. This choice aligns with prior con- figurations used in large-scale visual-language architectures (e.g., m0: 50 time-steps in Black et al. (2024), OpenVLA: 8 time-steps in Kim et al. (2025)). We also provide an ablation study on the context window, refer to Appendix D. Notably, our models are significantly smaller, yet demonstrate strong capability for long-term, temporally consistent predictions within this horizon as the results in Fig. 2 indicate.
4.2 Open Quantum System
To benchmark MetaSym on a novel open quantum dynamics task, we consider a parametric oscillator initial- ized in a coherent state (Gardiner & Zoller, 2010). The Hamiltonian includes a harmonic term A = wāt a, a squeezing term Algg = !x (at2 - a²), and a cubic drive Aeubie = 3(a3 + ata), with à, at the ladder op- erators in truncated Fock space of dimension N. Dissipation arises via coupling a thermal bath with rate 7 and mean occupation ūch. Continuous heterodyne detection yields a stochastic master equation for the conditional state (Appendix B.2). The only experimentally accessible data are the real and imaginary quadratures X and P of the heterodyne current, extracted from single-shot trajectories, as the quantum state itsclf remains inaccessible. In this context, quadratures (X and P) denote the non-commuting obscrv- ables, serving as the quantum optical analogues to classical position and momentum (Milburn, 1987). We simulate this setup by numerically solving the stochastic master equation (Johansson et al., 2012), varying mcasurement efficiency y, squeezing strength X: cubic drive 3 oscillator frequency w, bath occupation îun- and initial states. Training and out-of-disst distrributt inference details are in Appendix B.2, As shown in Fig. 3, MetaSym performs well in this strongly stochastic, non-classical regime.
8

---

Published in Transactions on Machine Learning Research (02/2020)
0.4
Actual crin
Test MSE
0.1
0.0 0.5
1.0 1.5 2.0
At 2.5
3.0 3.5
A
8
Test Set Systems
Figure 3: (Top) Time evolution of the two quadratures measured via heterodyne detection for a representa- tive quantum system the test set, characterized by a measurement efficiency (1 = 0.86) and measurement strength (x). The orange lines indicate the true measurement trajectorics, while the blue lines display the model'ss predictions (MetaSym). The close overlap between these trajectories highlights the model's effec- tiveness in capturing the underlying quantum dynamics from heterodyne signals. The context window of the model is 30 time-steps. (Bottom) Plots showing the mean squared error (MSE) of each of the predicted quadratures (dots) for five randomly selected test systems. The consistently low median errors (boxes) across all tested systems underscore the robustness and generalization capabilities of the model.
Table 1: Trajectory error and parameter count against SOTA baselines for three OOD datasets
Spring-mesh System
Quantum System
Quadrotor
Models
Param, Count
Traj. MSE (+t)
Param. Count
Traj.
MSE (#a)
Param. Count
Traj. MSE (+.)
DHNNa
3.M
32 468 (28.086)
N/A
N/A
3138
26.375 (9.160)
Transfornrr
3.2M
36.653 (15.618)
194
0.950 (0.239)
4680
34.584 (14.063)
Naive Baseline (MLP)
3.5M
323.199 (13.959)
(0.226)
(13.987)
QLSTM (Flurin et al., 2020)
N/A
N/A
244
0.891 (0.233)
N/A
N/A
Dissipative SymODEN
3.2M
32.051 (20.337)
N/A
N/A
3187
29.174 (9.517)
FNO
(24.223)
7.648 (0.411)
3336
27.932 (11.157)
iMODE
3.2M
21.858 (11.552)
7.667 (0.436)
3204
57.190 (20. 903)
MetaSym (ours)
2.9M
19.233 (15.678)
130
0.859 (0.215)
3036
25.889 (8.967)
4.3
Quadrotor
For our final benchmark, evaluate quadrotor, canonical yet challenging system due to its floating- base dynamics and discontinuous orientat ion representations (Sanyal & Roy, 2023; Zhou et al., 2019). The floating base induces internal covariate shifts during training, as phase-space magnitudes could vary across time. Regarding its orientation representation, Euler angles suffer from singularities, while quaternions impose a unit-norm constraint that is typically handled via loss penalties, often leading to local minima and unstable training. To address this, we represent the pose of the quadrotor (i.e., position and orientation) using the tangent space of the SE(3) group (Solà et al., 2021), which avoids the explicit enforcement of the unit norm constraint. MetaSym is pretrained using simulation data for variety of different systems described in detail in in Appendix B.3. Importantly, the pretrained network is then fine-tuned via the query- value weight adaptation technique introduced with the ActiveDecoder to real-world dataset (Mohajerin & Waslander, 2018). The results not. only demonstrale our method's efficacy and robust ness on real-world systems but also the adattadapt capability of MetaSym that effectively diminishes the sim-to-rea gap. Even though a detailed real-worid deployment analysis is beyond the scope of the current paper, we performed an additional study verify our model's inference time. The results indicate that MetaSym can run on an NVIDIA RTX4090 in 1.551 ms for its full 30 time-step context-window.

---

Published in Transactions on Machine Learning Research (02/2026)
L
2.5
[.[m]
2.5
[m]
Actual Prediction
0.0
0.0
20
60
20
40
[rad]
pitch [rad]
yaw [rad]
WWW
0-
-
-2
-2
-4
20
40
60
20
40
60
20
40
60
L.gmm/ss py [kgmmjss)
Pe.y [kgm' py [Agmils]]
[kgmt(s] Peggmm [s]
2.5
0.0
M
0.0
0.0
My
Mm/ MM hm
-2.5
-2.5
225
20
40
60
20
40
60
20
40
60
0.5
0.5
0.5
-0.5
-0.5
-0.5
20
60
20
Time [s]
Time [s]
Time [s]
Figure 4: Represents the translational and angular phase-space evolution of the quadrotor, with training generated using the Crocoddyl trajectory optimization package (Mastalli et al., 2020). Each training system is initialized with randomized imitial conditions and randomized terminal position over a 1.5s horizon (150 time-steps) with a context window of 30 time-steps. The displayed test. trajectories represent the evolution of a real-world quadrotor. The ground-truth test trajectory (orange line) overlaps with MetaSym's 's predictions (blue line) indicating the excellent predictive capabilities of our architecture. The MSE across 54 world test trajectories is 0.028 + 0.091 for a context-windor of 30 time-steps. Note we translate the quaternions to roll-pitch-yaw angles to have better interpretability of the results.
4.4 Benchmarks
We benchmarked MetaSym against three baselines, an autoregressive Transformer (Vaswani, 2017), a physics- informed dissipative Hamiltonian neural network (DHNN) (Greydanus et al., 2019), naive Multi-Layer Perceptron (MLP), the Dissipative SymODEN (Zhong et al., 2020), the Fourier neural operator (FNO) (Li et al., 2021), and iMODE (Ti et al., 2023). Both MetaSym and the Transformer perform autoregressive next time-step prediction, whereas the DHNN models the system's 's symplectic and dissipative gradients, relying on an integrator for trajectory unrolling. The MLP also predicts the next state directly. Unlike the others, the DHNN cannot predict quadrature measurements due to ill-defined derivatives involving complex Wiener terms (Milburn, 1987), and its training proved less stable than that of MetaSym and the Transformer. All models were evaluated in an autoregressive roll-out regime. As shown in Table 1, MetaSym consistently outperforms the alternatives in horizoorizon prediction accuracy, particularly when using a 30 time-step context window. It achieves lower error accumulation and requires fewer parameters, which is especially advantagcous for high-dimensional systems like spring-mesher The variance present across all models is altributed to the noise that we artificially added to each network's input in order to highlight MetaSym's robustness and real-world applicability. In quantum dynamics, where real-time prediction and control are constrained by hardware latency and memory (Reuer et al., 2023), MctaSym's compact architecture offers significant advantage over Transformer- based architectures (Vaidhyanathan et al., 2024).
5 Conclusion
In this work, we introduced MetaSym, a novel deep-learning framework that combines structure-preserving symplectic networks with an autoregressive, decoder equipped with meta-learning for modeling a wide- range of physical systems. The core idea rests on striking a balance between strong physical priors, namely the intrinsic symplectic structure of Hamiltonian mechanics, and the fleribility required to capture non- conservative effects and heterogeneous system parameters. Our experimental results across diverse domains,
10

---

Published in Transactions on Machine Learning Research (02/2026)
ranging from high-dimensional spring-mest dynamics to open-quantum systems and real-world quadrotor systems, demonstrated that. MetaSym outperforms SOTA baselines in both long-horizon accuracy and few- shot adaptation with smaller model sizes even for real-world systems.
The SymplecticEncoder approximates canonical flows while preserving key invariants, significantly mitigat- ing energy drift and ensuring robust long-term predictions. The encoder's invertíble design enforces time- reversal symmetry and reduces error accumulation. Meanwhile, the Active/eecder models departs from ideal Hamiltonian evolution through autoregressive prediction and meta-attention. The resulting architecture is computationally efficient, given that it does not require explicit numerical integration during inference, and through meta-learning, it readily adapts to system variations with minimal additional data. This approach offers a scalable and unified framework for high-fidelity physics modeling in realistic settings with provable nea-ssyplecctic properties (see Appendix C.4).
Additionally, while a substantial line of work has explored Mcta-PINNs and paramcter-aware operator learn- ing to generalize across varying physical coefficients exist (Penwarden et al., 2023; Wang et al., 2021). These methods effectively adapt to new PDE instances, they typically enforce physical consistency via 'soft' regular- ization terms in the loss function. This reliance on soft constraints often necessitates careful hyperparameter tuning of penalty weights and does not strictly guarantee the preservation of conservation laws during infer- ence. In contrast, MetaSym enforces symplectic symmetries through "hard' architectural constraints within the encoder. This design ensures that the core conservative dynamics remain on the symplectic manifold by construction, independent of optimization stability, while relegating only the non-conservative dynamics to the adaptable decoder.
Limitations & Future Work. It is important to note that MctaSym's strongest guarantees (e.g. bounded energy drift, near-symplectic behavior, and long-horizon stability) rely on the availability of at least some representative conservative data for pretraining the SymplecticEncoder. When such data are unavailable, MetaSym can be interpreted as a hybrid physics-informed / data-driven model whose guarantecs degrade continuously toward those of fully black-box architecture. Considering MetaSym's promising performance, we seek to investigate several future direct ions such as incorporating a fully symplectic network for modeling realistic physics by exploiting the underlying structure of dissipation and our control signals. MetaSym's 's effectiveness real-world system data also remains to be investigated. Another natural extension of few-shot adaplation is online learning and control, since MelaSym can quickly adapt to new system configurations with minimal data. This could be leveraged in real-time control loops and model-based Reinforcement Learning (RL) algs Finally, future research could focus on how to integrate the decoder's adaptation step with RL or adaptive Model Predictive Control (MPC) frameworks, effectively enabling self-tuning controllers in rapidly-changing environments.
References
Yuri Alexeev Marwa H. Farag, Taylor L. Patti, Mark E. Wolf, Natalia Ares, Alán Aspuru- Guzik, Si- mon C. Benjamin, Zhenyu Cai, Zohim Chandani, Federico Fedele, Nicholas Harrigan, Jin-Sung Kim, Elica Kyoseva, Justin G. Lietz, Tom Lubowe, Alexander McCaskey, Roger G. Melko, Kouhei Nakaji, Alberto Peruzzo, Sam Stanwyck, Norm M. Tubman, Hanrui Wang, and Timothy Costa. Artificial intelligence for quantum computing, 2024. URL https: //tt//aaavv///////////////aa// weaazii.rg///eeeee www.oogg///////eeeee//oooeeeee/e abs//411..111..////////////1/111 09131.
Natalia Ares. Machine learning as an enabler of qubit scalability. Nature Reviews Materials, 6(10):870-871, 2021.
Kaixin Bai, Lei Zhang, Zhaopeng Chen, Fang Wan, and Jianwei Zhang. Close the sim2rcal gap via physically- based structured light synthetic data simulation, 2024. URL https://araat httpa/:///xwvvooo//bbbbb org/abs/24000 12449.
Somil Bansal, Anayo K Akametalu, Frank J Jiang, Forrest Laine, and Claire J Tomlin. Learning quadrotor dynamics using neural network for flight control. In 2016 IEEE 55th Conference on Decision and Control (CDC), pp. 4653-4666 IEEE, 2016.
Johannes Bausch, Andrew W Senior, Francisco JH Heras, Thomas Edlich, Alex Davies, Michael Newman, Cody Jones, Kevin Satzinger, Murphy Yuezhen Niu, Sam Blackwell, al. Learning high-accuracy error decoding for quantum processors. Nature, pp. 1-7, 2024.
11

---

Published in Transactions on Machine Learning Research (02/2026)
Jacob Beck, Risto Vuorio, Evan Zheran Liu, Zheng Xiong, Luisa Zintgraf, Chelsea Finn, and Shimon White- son. A tutorial on meta-reinforcement learning, 2025. URL http::///arxi..or//ab/////// 2200000000000000088..
Petre Birlea, Toan Caşu, and Dan Comănescu. Optimization on the real symplectic group. Monalshefle für Mathematik, 191(3):465 485, January 2020. ISSN 1436-5081. doi: 10.1007//000005-0000-0100 attp://áie doi. www.i.org/.... 00000//000000000001886030
URL
Kevin Black, Noah Brown, Danny Driess, Adnan Esmail, Michael Equi, Chelsea Finn, Niccolo Fusai, Lachy Groom, Karol Hausman, Brian Ichter, Szymon Jakubczak, Tim Jones, Liyiming Ke, Sergey Levine, Adrian Ti-Bell, Mohith Mothukuri, Suraj Nair, Karl Pertsch, Lucy Xiaoyang Shi, James Tanner, Quan Vuong, Anna Walling, Haohuan Wang, and Ury Zhillisky. To: A lision-language-action flow model for general robot control, 2024. URL https: //axivv.// org/abs//////.oooooo httg/abb/2411.22//
Giuseppe Carleo, Ignacio Cirac, Kyle Crammer, Laurent Daudet, Maria Schuld, Naftali Tishby, Leslie Vogt- Maranto, and Lenka Zdeborová. Machine learning and the physical sciences. Rev. Mod. Phys., 91: 045002, Dec 2019. doi: 10.1103/Rew B000//ReeMoPPhs...o 0145000.. URL https://link.att apsnn.aapp..a a.s. RevModPhys. .1. 945000
Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud. Neural ordinary differential equations, 2019. URL attps://aras http::://axxvvorr///aaaaaa a0gg//bb/1800.....
Zhengdao Chen, Jianyu Zhang, Martin Arjovsky, and Léon Bottou. Symplectic recurrent neural networks, 2020. URL http:::///xxivvorgabbbeeee
13334.
Sang Kenn Choe, Sanket Vaibhav Mehta, Hwijeen Ahn, Willie Neiswanger, Pengtao Xie, Emma Strubell, and Eric Xing. Making scalable meta learning practical, 2023. URL
Miles Crammer, Sam Greydanus, Stephan Hoyer, Peter Battaglia, David Spergel, and Shirley Ho. Lagrangian neural networks, 2020. URL
Chelsca Finn, Picter Abbeel, and Sergey Levine. Model-agnostic meta-lcarning for fast adaptation of deep networks, 2017. URL https:
03400.
E. Flurin, L. S. Martin, S. Hacohen-Gourgy, and I. Siddiqi. Using a recurrent neural network to reconstruct quantum dynamics of a superconducting qubit from physical observations. Phys. Rev. X, 10:011006, Jan 2020. doi: 10.1103 MALL00/PPyyaeee..00.11..00 10.011006.
URL https://link.. aps
Crispin W. Gardiner and Peter Zoller. Quantum Noise: A Hondbook of Markovian and Non-Markovian Quan- lurn Stochastic Methods with Applications to Quantum Optics. Springer Series in Synergetics. Springer, Berlin Heidelberg, 3. ed., [nachdr.] edition, 2010. ISBN 978-3-540-22301-4
Valentin Gebhart, Raffaele Santagati, Antonio Andrea Gentile, Erik M Gauger, David Craig, Natalia Ares, Leonardo Banchi, Florian Marquardt, Luca Pezzè, and Cristian Bonato. Learning quantum systems. Nature Reviews Physics, 5(3):141-156, 2023.
Mathicu Geisert and Nicolas Mansard. Trajectory generation for quadrotor basod systems using numerical optimal control. In 2016 IEEE International Conference Robolics and Aulomation (ICRA), pp. 2958 2964, 2016. doi: 10. 10.1109/TCRA.20 TC.A.201..7777746
Nicholas Geneva and Nicholas Zabaras. Transformers for modeling physical systems. Neural Networks, 146: 272 289, February 2022, ISSN 0893-6080. doi: ne.tt.6/....eeuue.222222 www.eet.222..11.02.... URL http://dx.. doi, org/ 10. 1016/ neunet. 2021. 11. 0222.
Amin Ghadami and Bogdan I Epurcanu. Data-driven prediction in dynamical systems: recent developments. Philosophical Transactions of the Royal Society A, 380/2229):20210213, 2022.
Sam Greydanus, Misko Dzamba, and Jason Yosinski. Hamiltonian neural networks, 2019. URL https:
/ //arxiv.org wwwrzii.org/bbbe htrg/bbs/190/ 01t//199660.666.... 01563.
12

---

Published in Transactions on Machine Learning Research (02/2026)
Kurt Jacobs and Daniel A. Steck. A straighttorward introduction to continuous quantum measurement. Con- lemporary Physics, 47(5):279-300 303, September 2006. ISSN 1366-5812. doi: 10.100//001075100110093444 URL doi. ott.///0...000000000000000000000
Pengzhan Jin, Zhen Zhang, Aiqing Zhu, Yifa Tang, and Goorge Em Karniadakis. Sympnets: Intrinsic struclure-preserving symplectic networks for identifying hamiltonian systems, 2020. URL https://arxx: org/abs/2001.00 03750.
J.R. Johansson, P.D. Nation, and Franco Nori. Qutip: An open-source python framework for the dynamics of open quantum systems. Computer Physies Communications, 183(8):1760 1772, August 2012. ISSN 0010-4655. doi:
URL http://dx. doi org /0////1...0 1.t.//..1p.................. www.:ppc..........mm............ 2012. 02.12..2.0000 0200
Tom Kibble and Frank Berkshire. Classical mechamics. world scientific publishing company, 2004.
Moo Jin Kim, Chelsca Finn, and Percy Liang. Fino-tuning vision- Usion-languaag-acttio models: Optimizing speed and success. arXiv preprint arXiv:2502.19443 19645, 2025.
Yann A. LeCun, Léon Bottou, Genevieve B. Orr, and Klaus Robert Müller. Efficient backprop, pp. 9-48, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Springer Verlag, 2012. ISBN 9783642352881. doi: 10. 10.1077/97--3-622-552288 3. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.
Qiaofeng Li, Tianyi Wang, Vwani Roychowdhury, and M. K. Jawed. Metalearning generalizable dynamics from trajectories. Phys. Rev. Lett., 131:067301, Aug 2023. doi: MLLL03/PpysteeeLett.B...... URL https://link. aps. hetasoogg/////o// 11011100/PPPP www.0///yyaeeeeet....om 131.03301...
Weicheng Li, Lixin Zou, Min Tang, Qing Yu, Wanli Li, and Chenliang Li. Meta-lora: Memory-efficient sample reweighting for fine- tuning large language models. In Proceedings of the 31st International Conference on Computational Linguistics, pp. 8504 8517, 2025.
Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations, 2021. URL https://arxiv http:///araii.oorga//////00 org/abs//010.. 08895.
Daniel Manzano. A short introduction to the lindblad master equation. AIP Advances, 10(2), February 2020. ISSN 2158-3226. doi: 10.1063/ 100003/155115221 URL http://dz. doi org/r0..///// 106.111111111132
Carlos Mastalli, Rohan Budhiraja, Wolfgang Merkt, Guilhem Saurel, Bilal Hammoud, Maximilien Naveau, Justin Carpentier, Ludovie Righetti, Sethu Vijayakumar, and Nicolas Mansard. Crocoddyl: An Effi- cient and Versatile Framework for Catti-Contact Optimal Control. In IEEE International Conference on Robotics and Automation (ICRA), 2020.
G. J. Milburn. Quantun measurement theory of optical heterodyne detection. Phys. Rev. A, 36:5271-5277 Dec 1987. doi: ...110//Plyseev..3..... URL attps://lind: aps. org//or///. 1ttp://...11////heeeeeeee 36. 5271.
Nima Mohajerin and Steven L. Waslander. Multi-step prediction of dynamic systems with recurrent neural networks, 2018. URL
00526.
A Nichol. On first-order meta-learning algorithus. arXiv preprint arXie: 02999, 2018.
Yuji Okamoto and Ryosuke Kojima. Learning deep dissipative dynamics, 2024. URL https://arxiv.oo abs/2408. 11479.
Karl Otness, Arvi Gjoka, Joan Bruna, Danicle Panozzo, Benjamin Peherstorfer, Tesco Schneider, and Denis Zorin. An extensible benchmark suite for learning to simulate physical systems, 2021. URL https: war.ii.org/abb/lluuee 07799.
13

---

Published in Transactions on Machine Learning Research (02/2026)
Aristotelis Papatheodoron, Wolfgang Merkt, Alexander L. Mitchell, and Ioannis Havoutis. Momentum- aware trajectory optimisation using full-centroidal dynamics and implicit inverse kinematics. In 2024 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 11940 11947, 2024. doi:
108001774
Michael Penwarden, Shandian Zhe, Akil Narayan, and Robert M. Kirby. A metalcarning approach for physics- inrsics-ifforme neural networks (pinns): Application to parameterized pdes. J. Comput. Phys., 477( March 2023. ISSN 0021-9991. doi: 2..106//..ep.2223.1110 URL attps://do. org/10.101//1 http:::/6oo..rrg1..001////// jep. 2023. 111912.
Tobias Pfaff, Meire Fortunato, Alvaro Ganzhez- and Peter W. Battaglia. Learning mesh- based simulation with graph networks, 2021. URL https://arxiv. hwwlwworgg/abbo http//bbs/000.... 03409.
Aniruddh Raghu, Maithra Raghu, Samy Bengio, and Oriol Vinyals. Rapid learning or feature reuse? towards understanding the effectiveness of maml, 2020. URL https: //axxiv.// www.aiwww.waaaaaaa.a.. org/abs/1909 09157.
Aravind Rajeswaran, Chelsea Finn, Sham M Kakade, and Sergey Levine. Meta-learning with implicit gradients. Advances in neural information processing systems, 32, 2019.
Kevin Reuer, Jonas Landgraf, Thomas Fösel, James O'Sullivan, Liberto Beltrán, Abdulkadir Akin, Gra- ham J. Norris, Ants Remm, Michael Kerschbaum, Jean- Claude Besse, Florian Marquardt, Andreas Wall- ralf, and Christopher Eichler. Realizing a deep reinforcement learning agent for real-time quantum feed- back. Nature Communicaatinss, 14(1), November 2023. ISSN 2041-1723. doi: 10.1088/s41166:002--490000
URL http:/ //dx:://xx./ doi 000/10..11110000 1038/ 100000001////77 020000002222------
Antoine Royer. Wigner function in liouville space: canonical formalism. Physical Review A, 43(1):44 1991.
Sourav Sanyal and Kaushik Roy. Ramp-net: A robust adaptive mpe. for quadrotors via physics-informed neural network, 2023. URL https//////xxw.oo///aaaass 00000/90b22209099555
Lucas Schorling, Pranav Vaidhyanathan, Jonas Schuff, Miguel J. Carballido, Dominik Zumbühl, Gerard Milburn, Florian Marquardt, Jakob Foerster, Michael A. Osborne, and Natalia Ares. Meta-learning characteristics and dynamics of quantum systems, 2025. URL https: htpps://arxxvv http://awwivvoooaaaaaaaaaaaa
wrg/ 10492.
Joan Solà, Jeremie Deray, and Dinesh Atchuthan. micro lie theory for state estimation in robotics, 2021.
URL httpp::/arrti httpp/////axii.orrr//////112222/ org/abs/1812. 01537.
Andrew Sosanya and Sam Greydanus. Dissipative hamiltonian neural networks: Learning dissipative and conscrvative dynamics separately, 2022. URL https://arxiv. www.orrgg//bb////////////// 10085.
S.H. Strogatz. Nonlinear Dynamics and Chaos: With Applications Physics, Biology, Chemistry, and Engineering. Springer Series in Synergetics. CRC Press, 2nd ed. edition, 2015.
Pranav Vaidhyanathan, Florian Marquardt, Mark T. Mitchison, and Natalia Ares Quantum feodback control with a transformer neural network architecture, 2024. CRL bettps///rxxv..oom wttp////////.....o//./// 19253..
A Vaswani. Attention is all you need. Advances in Neural Information Processing Systems, 2017.
Sifan Wang, Hanwen Wang, and Paris Perdikaris. Learning the solution operator of parametric partial differential equations with physics- informed deeponets. Science advances, 7(40)=eabi8605, 2021.
Xuan Zhang, Limei Wang, Jacob Helwig, Youzhi Luo, Cong Fu, Yaochen Xie, Meng Liu, Yuchao Lin, Zhao Xu, Keqiang Yan, Keir Adams, Maurice Weiler, Xiner Li, Tianfan Fu, Yucheng Wang, Haiyang Yu, YuQing Xie, Xiang Fu, Alex Strasser, Shenglong Xu, Yi Liu, Yuanqi Du, Alexandra Saxton, Hongyi Ling, Hamnah Lawrence, Hannes Stärk, Shurui Gui, Carl Edwards, Nicholas Gao, Adriana Ladera, Tailin Wu, Elyssa F. Hofgard, Aria Mansouri Tehrani, Rui Wang, Ameya Daigavane, Montgomery Bohde, Jerry Kurtin, Qian Huang, Tuong Phung, Minkai Xu, Chaitanya K. Joshi, Simon V. Mathis, Kamyar Azizzadenosheli, Ada Fang, Alán Aspuru-Guus Guzik, Erik Bekkers, Michael Bronstein, Marinka Zitnik, Anima Anandkumar, Stefano Ermon, Pietro Liò, Rose Yu, Stephan Günnemann, Jure Leskovee, Heng Ji, Jimeng Sun, Regina Barzilay,
14

---

Published in Transactions on Machine Learning Research (02/2026)
Tommi Jaakkola, Connor W. Coley, Xiaoning Qian, Xiaofeng Qian, Tess Smidt, and Shuiwang Ji. Artificial intelligence for science in quantam, atomistic, and continuum systems, 2024. URL https://arxiv.orrs org/ 2ha//2377008423.
Yaofeng Desmond Zhong, Biswadip Dey, and Amit Chakraborty. Dissipative symoden: Encoding hamiltonian dynamics with dissipation and control into deep learning, 2020. URL https://arxiv. http://v..r//bb/////////// 08860.
Yaofeng Desmond Zhong, Biswadip Dey, and Amit Chakraborty. Symplectic ode-net: Learning hamiltonian dynamics with control, 2024. URL btttp://aaxxiv.orggbbb... 190971
Yi Zhou, Connelly Barnes, Jingwan Lu, Jimei Yang, and Hao Li. On the continuity of rotation representalions in neural networks. In 2019 IEEE/CVF Conference on Computer Vision and Pottern Recognition (CVPR), 2019. doi: 11109//VVR.20111
pp.
Luisa Zintgraf, Kyriacos Shiarli, Vitaly Kurin, Katja Hofmann, and Shimon Whiteson. Fast context adapta- tion via meta-learning. In International Conference on Machine Learning, pp. 7693 7702. PMLR, 2019a.
Luisa M Zintgraf, Kyriacos Shiarlis, Vitaly Kurin, Katja Hofmamn, and Shimon Whiteson. Fast context adaplation via meta-learning, 2019b. URL / wttp://w.wr//b//./...ooo 03642.
15

---

Published in Transactions on Machine Learning Research (02/2026)
A Meta Learning Setup
As mentioned in Section 3, we claborate on the algorithmic pipeline of our SymplecticEncoder and ActiveDe- coder modules. In particular, we outline the MAML-based mela-learning stralegy for the CompleclicEncodeee and the bi-level adaptation meta-learning for the AcliveDecoder Fig. 8 provides an interpretable insight into the effect of inner adaptation during the meta-upa ceta-update step and its effectiveness.
A.1 Encoder
Following Scction 3.1, Algorithm provides a high-level overview of how the SymplecticEncoder is trained via MAML meta-learning. More specifically, the Model-Agnostic Meta-Learning (MAML) is a meta-learning framework designed to train models that can adapt quickly new tasks using only small amount of data. The core idea is to learn an set of parameters that is not task-specific, but task agnostic. As a model- agnostic method, it can be applied to any model trained with gradient descent, including neural networks case with MetaSym), reinforcement learning, and beyond. During
for classification, regression (e.g., meta-training, MAML simulates adaptation by sampling systems, performing inner-loop updates on each system, and then optimizing the initial parameters via meta-grade greaingrsdeents computed across systems. This results in a model that learns how to learn, making it highly effective in few-shot scenarios where fast generalization from limited supervision required.
The task-specific gradient steps are performed on parameters that initialized from parreedee set, denoted as 0sg, for each system. These inner-loop (fast adaptation) updates are applied to independent copics of the initial parameters, denoted / one for each task (i.e., each system i). Crucially, the original OsE must remain unchanged during these lask-specifie updates, as it serves as the point from which all adaptations are made.
Only after the fast adaptation steps across tasks are completed do compute the meta-gradient, based the post-adaptation performance on each task, and use it to update the original initialization OSE via the outer-loop (meta) update. This structure ensures that Osg is optimized for adaptability, enabling efficient transfer to unseen systems.
16

---

Published in Transactions on Machine Learning Research (02/2026)
Algorithm Meta-Learning for the SymplecticBhcoder
(D()) ..... D(N}; Require: (D......,.N... Training data from related systems.
Each D(1) is trajectory
(qt) Pt 4
(9) T )
1: for epoch = to
N epochs do for each mini-batch of systems B C (1....,NN N) do optimizer theta. zeroaareerrrdddlooo inner inner_optiine cecer_optimizertte _eroo_grad for each system i E B do
3:
4:
Split the trajectory D(i) into:
Indapt C (1, (.....,T(())) Tadapth:
Imeta = (1,..., Tel Indapt-
7:
Lmeta + Fast Adaptation: Adapt system-specific parameters
10:
g +- fgg. clone() ()tach for 1 to K do
b. Detach
11:
12:
inner_ aimeroopimiize.... zero zero_grad ((t) +
13:
14:
for t E Indapt do
15:
(9 P++1) ((+11PP411. t 4 ese
(2)
9: P
(1)
u
(
SE
16:
c inner -((
innner 19 9
P
(0)
1941
P
(8)
17:
end for
(Cloner/ Tadaptt backwardd()))
18:
Update 0 only
19:
inner ceter__ptimizer.ee step))
20:
end for
21:
Meta Update: Update the global parameters 8g
22:
for t E Imeta do
23:
OSE - ()) c)
D No detach
24:
(9 1141) t @
q
(2)
ASE
25:
Litaa & Smeta
[9.
P+11
(8)
19
P 1
(4)
26:
end for end for
27:
28:
(Lmeta/ (T( (ackward
Eq. 4
29:
theta.step
> Update OsE only
30:
end for
31:
end for
A.2 Decoder
Algorithm 2 provides high-level overview how the Active is trained via meta-learning.
17

---

(8) Published in Transactions on Machine Learning Research (02/2026)
Algorithm 2 Meta-Learning for the AclideDecoder
Require: D= . Di Training data from N related systems. Each D(4) a trajectory of states
(qt) Pt u
99) ( t=1"
1: for epoch = to
N epochs do for each mini-batch of systems B S (1....,N] do
3:
optimizer @erimizer_theta... theta.zero_at zero_eero__eeee _orrd_) for each system E B do
4:
Split the trajectory D(i) into:
Tadapt C (1,..., T-()) addpth Imeta = (1,..., T Tuetaa Tadapt-
6:
7:
Inner loop: Adapt local parameters Ç for k ==1 to K do
for t E Zadapt do inner_ comner_ptimizer..om zero (radd( G:randomize()
10:
11:
Reinitialize Si
12:
(9++1) P 1+1 t AAD Uhe
Pri
tot
OAD Çi
13:
c6) inner + 1 P 4et
(0)
P
(8)
14:
/ / eddpt) backward
D Eq. 3.2
15:
inner inner_optimizeee www.oppptii....... step()
Update G only
16:
end for end for
17:
18:
Outer loop: Meta-update for the global parameters OAD optimizer_ther theta. zero_grade (rrdd)
19:
D. No update to Ç now
c +
20:
21:
for 1. E Imeta do
(9 [(1]
(7)
(i)
(1)
22:
PAAn Atce Piet
u AAD Ç
(()) outer Louter + 111 1 Pit1 14+1 end for (6) (
(8)
(0
Pi-1 11
23:
24:
25:
meta backward()
④ Eq. 3.2
26:
optimizer teee_thtttttee () ()
> Update AD only
27:
end for
28: end for
29:
end for
B Experimental Setup
B.1 Spring-mesh System
Spring networks are simple proxy for numerous physical scenarios involving deformable solids and cloth (in computer graphies, mechanics, or robotics). Each pair of connected particles exchanges spring forces that depend on displacements from rest lengths. Viscous-damping terms further shape the evolution. While the individual dynamics (Hooke's law) are straightforward, the combination of hundreds of coupled springs can give rise to complex large-scale deformations and oscillations. By benchmarking on spring-mosh we can examine how MetaSym learns large deformations, wave propagation through membrane, the impact of damping.
The training datasct consists of 25 distinct spring-mesh systems, each characterized by unique set of physical parameters, including spring stiffness, damping coefficients, and initial conditions as indicated by Table 2a. These parameters are sampled from predefined distribution to ensure sufficient diversity within the training set. Each system is simulated over time span of 2000 irregular time-stops, capturing the full trajectory of node displacements and momenta. The resulting dataset provides a rich representation of dynamical behaviors within the parameter space.
18

---

Published in Transactions on Machine Learning Research (02/2026)
To assess generalization and robustness, construct test dataset comprising ten additional spring-mes mesh systems. Unlike the training set, the parameters for these systems are drawn from distributions that differ from those used during training as indicated by Table 2b, introducing a domain shift that mimics real-world variations. This OOD test set enables a rigorous evaluation of the model's ability to extrapolate beyond the observed training dynamics and adapt to unseen conditions. For further information regarding the spring-mesh benchmark refer to Otness al. (2021).
By incorporating both in-distribution training data and OOD validation data, this experimental setup ensures comprehensive assessment of the model's learning capacity, robustness, and generalization performance when applied to novel physical configurations.
Table 2: Parameter ranges for in-distribution and out-of-distribution regimes.
(a) In-distribution parameters T 2000
(b)
Out-of-distribationn parameters T = 2000
Parameters
Values
Parameters
Values
Vidccyy kg/s)
Uniform 0.1, 0.2)
"decay
kg/s)
Uniform (0.001 0.05)
mass kg]
Uniform 00.1 , 2.0)
mass kg
Uniform 5.0)
K Kspring N/m
Uniform(O.00 0.5)
Kspring N/m
Uniform(1 (1.0, 3.0)
Init. Conds. m
Uniform (0, 0.6)
Init. Conds. m
Uniform (0.9, 2.5)
dt S
Uniform (0.001, 0.03)
dt [s]
Uniform (0.1, 0.3)
Mesh Size [g ngy
10 x 10
Mosh Size [2 ny
10
B.2 Open Quantum System Derivation
Understanding the behavior of quantum systems plays a vital role in the development of promising technolo- gies such as quantum computing, metrology, and sensing. While deep-learning has found great success in several areas such as quantum control (Vaidhyanathan et al., 2024), error corroction (Bausch ct al., 2024) and luning (Ares, 2021; Gebhart et al., 2023), predicting the measurement record based on modeling quantum dynamics has long remained elusive. In many scenarios, the system of interest is open: it couples to en- vironment (or bath) that can introduce thermal noisc and dissipation. Furthermore, contimuous monitoring (e.g., via homodyne detection) adds additional measurement backaction, reflecting fundamental constraints from quantum mechanics (Jacobs & Steck, 2006). Capturing these noise and measurement effects is pivotal for accurately predicting quantum trajectories and devising robust control protocols.
Unlike closed Hamiltonian evolutions, open quantum systems require one to solve Stochastic Master Equa- tions (SMEs) incorporating decoherence and measurement, terms. These equations produce trajectories of the (mixed) quantum state conditioned the noisy measurement record. In many practical settings, however, only have direct access to certain observables (e.g., position and momentum quadratures) rather than the full quantum state. Hence, training deep learning network to model the quantum system and to predict future measurement outcomes becomes a natural and practically relevant challenge. The SME describes the evolution of the conditioned quantun state (t) under the effect of environmental and measurement noise as in Jacobs & Steck (2006)
dt.(t) = -ii po(H) Eplbslo pppl(Mppp((a
(7)
- LpLt where the dissipator is D that encodes coupling to the euvironment (e.g., photon loss to reservoir, thermal excitations, dephasing, etc.) (Manzano, 2020). The stochastic backaction term, H[MP = MptpMi- [M p) p. describes +
p) ( Each Linblad operator, L,, represents a collopse operator
continuous monitoring of an observable M, with n the measurement efficiency (0 S n s 1). The Wiener increment, dW,, captures the randomness inherent in quantum measurement outcomes.
19

---

Published in Transactions on Machine Learning Research (02/2026)
B.2.1 Setup for Parameteric Oscillator
In Section 4.2, we focus on a single-mode bosonic system with annihilation operator à and creation operator at The Hamiltonian is:
H = wäta + ix 3 2 + +
(8)
with w, X and / being the oscillator frequency, squeezing strength and cubic driving term respectively. We also include two Linblad operators,
L = V (Teh + 1)ā, L Vmnan',
(9)
where y is the coupling rate to the thermal bath, and ath is the average occupation number.
Quantum Trajectory Confidence Interval 0.75, 0 0.8
Single Nolay Trajectory
Unsemble Mean (100
Thaoretical Mean Eq)
Confidence Interval (m)o)
Expectation 60
Time
Figure 5: Quantum System's position quadrature X evolution: The faint gray traces represent individual single-shot trajectories, showing the inherent randomness caused by mcasurement backaction and thermal noise, directly visualizing the noisy nature of the dataset. The blue shaded region marks the confidence interval, quantifying the standard deviation the quantum state's fluctuations over time. This demonstrates the jaggod unprodictability of individual runs. The ensemble average (solid blue line) converges the smooth, and deterministic prediction of the Master Equation (red dashed line) over multiple trajectories.
A common measurement technique often employed in experimental settings is called heterodyne measure- ment. Heterodyne detection continuously measures both quadratures of the output of our dissipative system by mixing it with a local oscillator at a slightly shifted frequency and then demodulating the resulting beat signal. This yields two simultancous photocurrents, often referred to as the in-phase (T) and quadrature (Q) components. This reflects a key practical outcome of solving Eq. is that po(t) depends on this random measurement trajectory.
We employ heterodyne detection of the field operator à. Based on this measurement scheme, we can get separate the real and imaginary parts to obtain quadrature values that roughly correspond to X and P while adding quantum noise and measurement uncertainties due to quantum mechanical effect (Flurin et al., 2020).
In the following Tables 3b & 5a, we describe the parameters used generate our dataset by solving the SME in order to generate training data.
20

---

Published in Transactions on Machine Learning Research (02/2026)
Table 3: Parameter ranges used in training and evaluation of quantum trajectories. The in-distribution set represents conditions seen during training, while the out-of-distribution set introduces significant shifts in frequency, squeezing, thermal noise, and measurement efficiency.
(a) In-distribution parameters. Time step dt 0.5, total (b) Out-of-distribution parameters. Time step dt 0.5, duration T = 600.
total duration 7 = 600.
Parameter
Value
Parameter
Value
oscillator frequency Uniform(0055, 1.0)
oscillator frequency
Un.1orm(0.1,0.4)
squeezing strength
Uniform (0.1, 0.4)
squeezing strength
Uniform (0.5, 0.8)
X
X
thermal
occup. Uniform (0.1 0.5)
thermal occup.
Uniform 0.7)
(na)
(men)
meas. efficiency 1
Uniform (0.7,
meas. efficiency
Uniform (0.44, 0.6)
We also include, Fig. 5, that illustrates the stochastic evolution of the position quadrature X for our system. The nature of the system's stochasticity is revealed by the shaded confidence interval, the shape of which is dictated by Eq. 7.
B.3 Quadrotor
The quadrotor system challenges data-driven methods with its floating-base dynamics. To generate the train- ing and OOD validation datasets use Crocoddyl trajectory optimization package based the dynamics model proposed by Geisert & Mansard (2016). The training dataset comprises 30 systems with randomized parameters such as inertia, torque constant and rotor lengths as indicated in Table 4a. Each trajectory is generated from a randomized initial condition to a random terminal position with zero velocity, within a pre-set bounding box, to avoid unrealistic velocities.
In the same manner the OOD validation set contains 10 trajectories, each one corresponding to system with parameters drawn from the distributions indicated in Table 4b.
Table 4: Parameter distributions for simulating quadrotor dynamics. The in-distribution set covers training conditions, while the out-of-distribution set reflects variations in inertia, mass, center of gravity shift, and torque coupling beyond the training domain.
(a) In-distribution parameters. Time step dt 0.01, duration T' = 250.
(b) Out-of-distribution parameters. Time step dt
0.01, duration T' 500.
Parameter
Value
Parameter
Value
Inertia Iaiag
Uniform (0.1, 0.2)
Inertia Zdiag
Uniforml (0.3, 0.7)
mass m
Uniform (0.5, 3.0)
mass m
Uniformi (3.0, 5.0)
deogs
Uniform (0.2, 0.5)
deogs
Uniform (0.5, 0.7)
Crorques
Uniform (0.001, 0.1)
Crorques
Uniform (0.1, 0.2)
The results in Fig. verify the accuracy and effectiveness of our method for simulated floating-base dynam- ics systems. This consists the pre-training executed before fine-tunin the architecture on the real-world quadrotor dataset illustrated in the main text.
21

---

Published in Transactions on Machine Learning Research (02/2026)
d. [m]
4y [m]
E
7.5
5.0
-2
0.0
1.0
-2
0.0
1.0
2.5
0.0
1.0
7
p
Median(q)
Actual
Prediction
6
Median(p)
58, [rad]
58, [rad]
60, [rad]
2.5
5
0.0
0.0
1.0
0.0
1.0
0.0
1.0
Test MSE
4
Ps [kg m/s]
[kg m/s]
P. lkg m/s]
3
2.5
2.5
0.0
0.0
-2.5
-2.5
2
0.0
1.0
0.0
1.0
Pe.x [kg rad/s]
Pe.y [kg rad/s]
Po,z [kg rad/s]
2
2
2
0
0
0
-2
-2
-2
-1
1.0 Time [s]
0.0
1.0
C D E
Time [s]
Time [s]
Test Set Systems
Figure 6: (Left) Represents the translational and angular phase-space evolution of the quadrotor, with training and test trajectorics generated using the Crocoddyl trajectory optimization package (Mastalli al., 2020). Each task is initialized with randomized initial conditions and randomized terminal position over 1.5s horizon (150 time-steps) with context window of 30 time-steps. The ground-truth test trajectory (orange line) overlaps with MctaSym's predictions (blue line) indicating the excellent prodictive capabilites of our model. (Right) Plots summarizing the mean squared error (MSE) of each of the phase-space coordinates evolution for five randomly generated test systems (dots). The consistently low median errors (boxes) across all components underscore the robustness and generalization capabilities of the model.
C (p)) Near-Symplectic Architecture
In this section, prove that composing our Symplecclicreeuder Pasz with our AeliveDecoder tEAD handling control signals and dissipative effects yields a near-symplectic map, with an explicit bound on preserving approximate symplectic geometry even when tested in adverse dissipative conditions. This formalizes the realistic effects we can introduce to MetaSym and the inductive bias provided by symplectic invariants.
C.1
Overall Map
The full transformation is
(qr, P:) + = Doyn (qr, + za DoAD (Ze: Ut de).
Hence, at the global level, define
l Pt, Ur, de) = Po (Boss ( (qe, Pt), us de).
We aim to show that if LEAD remains a small (bounded) perturbation O(p) from identity in sp-ce, then @ preserves the symplectic structure up to small error term. This is called O(p) near-spmplecticity, We assume that the dissipation d, and control-input u are separable for the sake of this proof.
C.2 SymplecticEncoder
From Jin al. (2020), we know that LA- NetSypsNess are fully symplectic due to their construction. By extension, we can show that Synpllccticnncoder Dogg is symplectic.
22

---

Published in Transactions on Machine Learning Research (02/2026)
Definition C.1 (Symplectic Property). Let X Rddd represent the canonical phase space with coordinates x (q.P). We write
Ia
J
Id 0 0
the standard 2d x 2d symplectic matrix. A differentiable map V X + X is striclly symplectic if
( JdV((x J V X. This implics dct(d (volume preservation).
We know
PosR X + Z S Rdd
is strictly symplectic, i.e.
d (x) T J Jd (x)= J, del(d (x)) =
Internally, since PosR is composition of symplectic sub-blocks (c.g. LA-SympNets). Its parameters, OSE. can be partially meta-learned as long as the strict symplectic property is relained.
C.3 Decoder with Control and Dissipation
We define
S OAD
2XUXD + 2,
where:
2 C 2d is the latent phase-space output by (Doys);
U I represents control signals (bounded by |uu/|| S U Umax);
D C IR' represents dissipative parameters (bounded by ||d,] || < Dmax), which model forces that remove or drain energy (e.g., friction drag).
In order to account for the effects of the cross attention, the decoder modifics the latent state by
R (Ze: Uz d.) 2. FOAD (Ze: U, de);
where (FeAD) can cross-attention with magnitude modulated by ||u/ll, or a damping formula modulated by |d.l].
C.4 (pp) Syapletcpecci Proof and Explicit Bound
In this section, we prove that z, + Ze + FeAD is a bounded perturbation from identity (in partial derivatives), then the composition with a strict symplectic map remains close preserving J.
Bounded Perturbation Assumption. For the ActiveDecoder map, we take the partial derivative w.r.t. in FoAD is bounded by linear-type function of (Iluell l[del)):
OFeAm 8%.
(Ze: u, de) < oo au |luell oua ||dell:
for some constants ag, Ouy ag 2 0. This covers cross-on cross-attentnn attention scaled by ||u ll and dissipative scaled by ||d
terms. Since ||eu|| < Unax and | < Dmax: we define:
= ag + an Umax + ad Dmax.
Hence, max un d. |/ DOAD I|l < p. Equivalently,
dd. Feao (Ze, us, dr) S V(zoo ur
23

---

Published in Transactions on Machine Learning Research (02/2026)
The Composed Map. Recall define the global map
L Pr, Ur de) = DoaD Dese (qr, ur de).
Writing = (q:; Pt) E 2d2d for convenience, we have
D8sr (x) and Za = Poso(Zes us de).
To show near-symplecttcity,, we study
d u, d1) = dz. Doan Ze. ue di) x
do (x)
I+A, ||
strictly symplectic.
Theorem C.1 (O(p) Near-Symplectic Composition). Suppose PasE is strictly symplectic, i.e.
d (x) J d (x) J and det(d@oss (x)) =.
Also assumee dz LOAD satisfies the doundedd-peruuraatiin condition max l/ DRAD -1ll S p over ||e|l S U Umx d S Dmaxx Then for the composed map Lo, we have:
(x) J clPo(x) - J/| s Cp.
for constant C > 0 depending on the norm of d (x)) Hence & is esymplectic with e = C.P. C.p. Further- more,
(ct(d(pg -
implying volume preservation as well.
Proof. Let (qu.P)), .BSs (x), and I A dg, dee (Ze, ue de) with ||||| s p. Then
d uc de) (I + A) d AsE (x).
Hence
d J d = d (I+ A) J (I+ + A) ddeSE
Expanding AI+ A) J ATJ JA + A JA + O(I) we substitute d& Osg
J Jd J:
d Jd = J + 0(1 = + O(p).
In operator norm, ||d Jd J| S Cpp. Since d is volume-presorving, dct(d = det([ A) = 1 + Opp..
Explicit Cross-Aue Attestiitnn Bound. For instance, if FeAD includes a cross-attention term scaled by ||u|]] plus a dissipative term scaled by ||d/]|, we might write
FeAD (Zer Uy de) = a CrossAun((ee u.) - (d]z.),
where d n. indicates some parametric dissipator. Tf CrossAttn has partial derivative z, normed by ||u,]]
and d, b linear in [[dr],, then
FFOAD 8hc
S og au ||ue|| + aa dell,
giving the same p= og + Cg Unax + aaa maax-
We choose not to explicitly bound this perturbation during training to allow for MetaSym to model extremely dissipative systems. In Appendix D.2, we provide the perturbation bound Cp empirically for systems un- dergoing extreme dissipation and control. We observe that this bound is 1 even under these extreme settings.
24

---

Published in Transactions on Machine Learning Research (02/2026)
D
Ablation Studies
To rigorously assess the efficacy of our proposed framework relative to currett SOTA methods, we conduct a series of ablation studics. All experiments are performed on a moderately complex yet tractable spring-mas mesh system consisting of a 3 x node grid. Unless otherwise noted, all other hyperparameters remain identical those Tables 2b.
D.1 Modeling Conservative Systems
The SymplectieEncoder is designed explicitly preserve symplectic structure by learning a generalized latent, symplectic basis, which captures the conservative phase-space dynamics. By design, it conserves invariant quantities such energy and volume in phase space (Jin et al., 2020). The AcliveDecoder models cestem-parameter variations and captures dissipative and control-related effects as bounded perturbations to the symplectic manifold (sce Section C.4). Hence the architecture is equipped to model fully-conscrvative systems and adapt to changing parameters. We conduct two different experiments in order to verify the efficacy our architecture and the validity of our claims.
D.1.1 Conservative Spring-Mest Mesh System
We test our architecture against conservative spring-mesh at test time. Specifically, the ActiveDecode has been pre-trained on dissipative data and we adapt it using our meta-learning procedure to 25 fully conservative spring meshes, The achieved trajectory MSE across these systems is 0.050 + 0.016. This demonstrates that our meta-learning paradigm can accommodate substantial variations in system paramcters through the ActiveDecoder. Simultaneously, the architecture respects the relevant physical invariances.
D.1.2 Conservative Harmonic Oscillator
A harmonic oscillator is the prototypical example taught in many undergraduate courses in dynamics. It constitutes a well-studied system with analytical solutions. Stability analysis and dynamics' research (Stro- gatz, 2015) has equipped with the tools necessary to study these linear systems in depth. By extension, understand further architecture's true predictive capabilities in terms of physical invariances, such energy and phase-space volune conservation.
The
Equation (ODE) modelling the harmonic oscillator's dynamics in phase-space coordinates is considered in Eq. 10,
d (x(1) di p(t)
1
x(1) p(t)
mmw
(10)
where w V is the oscillator frequency with mass and spring constant k. The position is indicated by xtt) and the momentum coordinate by p(t). The analytical solution for this is given by Eq. 11,
(x(t) p(()
(
cos(wit) sinwt)
miw sin(wt) st((aat)) cos(wr)
(11)
where (oo.P)) po) are the initial conditions.
We Irain MetaSym with harmonic oscillators with varying masses and spring constants depicted in Table 5b. The results in terms of phase-space trajectories, energy-conservation are highlighted Fig. 7.
Beyond energy conservation, Hamiltonian dynamics preserve phaso-space volume according to Liouville's theorem. For the one-dimensional harmonic oscillator, this implies that the area enclosed by any closed orbit in the (I.,) phaso-space must remain invariant over time. To quantitatively assess whether MetaSym respects this structural property, compute the phase-space area enclosed by both the ground-truth and predicted trajectories using a discrete line-integral (shoelace) estimator. More specifically, the enclosed phase-
space area of a closed discrete trajectory N is computed as AN = 1 (ga441 q++pp)) In
25

---

Published in Transactions on Machine Learning Research (02/2026)
Atuut Predition
-10
-20
m. []
Time [4]
(a) Time-evolution
10
Aedimtinn
Mechanical Eiergy
-4-
x [m]
Time [s]
(c) Phaso-space trajectory
(b) Energy Conservation
Figure 7: (a) MetaSym predicts the dynamics of the harmonic oscillator with high accuracy, since the prediction overlaps with the ground truth. (b) The energy drift of MetaSym during autoregressive rollouts remains negligible and uniformly bounded, as evidenced by the small, repetitive oscillations observed over time rathor than any sccular growth. (c) The overlapping prediction and ground-truth closod-orbits further support our claims of physical-invariance preservation.
the continuum limit N oo, this converges to the symplectic line integral A pp Jf dg A dp, corresponding to the symplectic area enclosed by the orbit.
Table 5: Training and out-of-distribution testing paramcter ranges for the harmonic oscillator.
(a) In-distribution parameters. Time step dt 0.01, total (b) Out-of-distributiio parameters. Time step dt 0.01,
duration 7 = 3000.
total duration 800.
Parameter
Value
Parameter
Value
Mass (m (kg))
Uniform 1.0)
Mass (m [kg))
Uniform (2.5, 3.0)
Spring Const. (k [N/m)
Uniform 4.0)
Spring Const. (k /m/)
Uniform (5.0, 6.0)
Init. Pos. (3o (m))
Uniform( 1.0, 1.0)
Init. Pos. (ro [m])
Uniform (1.0, 1.5)
Init. Vel. (Uo mm/s)
Uniform(-1.. 1.0)
Init. Vel. (wo m/s))
Uniform (-1.5, (-1.5,-1.00 -1.0)
Concretely, for the representative system, the true enclosed area. is Atrue 21.602, while MetaSym predicts Apred 21.619, corresponding to a relative deviation of less than 0.08%. This close agreement indicates that MetaSym not only reproduces the correct qualitative closed-orbit geometry (Fig. 7c), but also preserves phase-space volume high accuracy during autoregressive rollouts. Such behavior consistent. with an approximately symplectic evolution and supports the claim that MetaSym captures fundamental geometric invariants of conservative dynamics beyond pointwise trajectory fitting.
D.2 Empirical Perturbation Bound Estimation
The ActiveDecoder models the dissipative and control-input effects of system perturbations the symplectic manifold that the conservative dynamics of given system evolve While Section C.1 provides
26

---

Published in Transactions on Machine Learning Research (02/2026)
definitive argument in favor of this interpretation, calculating this bound analytically is not possible. To further support our claims we performed an ablation study that calculates this bound empirically, using the pretrained weights for dissipative 3x3 spring- mesh system with parameters such as decay that are chosen to represent a real deformable surface such as polypropylene undergoing extreme deformation (eg. a trampoline). In this edge case, The bound Cp is calculated as below:
||d - J|l2 = 0.999
(12)
La
where J
and &g.- the trained network.
IId 0
D.3 Effectiveness of Meta- Attention Mechanism
Our meta-learning framework adapts the Query and Value projections of the cross-allention in the ActiveDe- coder, enabling task-specific modulation of decoded representations. As shown in Table 6, this meta-attention approach consistently outperforms MctaSym without meta-attention as well as standard pre-training and fine-tuning, across three dynamical systems with realistic dissipation and inertial parameters.
Table 6: Meta-learning performance (MSE + std) across OOD systems.
3x3 Spring-mesh
Quantum System
Quadrotor
ActiveDecoder finetuning w/o meta-attention with meta-attention
0.405 1 0.536
1.068 1 0.165
8.814 + 3.328
0.613 + 0.432
1.000 = 0.200
16.457 + 15.184
0.230 0.115
0.898 = 0.241
8.397 5.906
Meta-learning shapes the parameter initialization such that gradients on small adaptation set are more informative and better aligned with task-specific directions, enabling rapid specialization and improved training dynamics during adaptation, As result, the model achieves lower MSE across all systems despite limited adaptation data, consistent with prior theoretical and empirical findings on gradient- meta- learning lew-shot regimes (Raghu al., 2020; Beck et al., 2025). In this ablation, we consider common nominal values for all system parameters in order to focus on the efficacy of the meta-learning mechanism rather than robustness to high-noise scttings.
Finally, as showcased by Fig. 8 for the quadrotor, the fest-adaptation achieves smooth convergence as the overall training converges with significant loss minimization. This proves the smooth dynamics of our meta- learning framework and the absence of severe instabilities that could have hindered its performance.
Inner Adaptation
Inner
30002
(0:2
Syx
21:21
5s.
00:2
Scaled adaptation
23000
9ys (0))
5p. 10:1
Sys 10:4
59. 1211
9ys 0.17
Syt 1.
3ss. 0:25
9ys 20-2
1ys 26
Scaled adaptation
30000
17
20
Sp. 10:8
59 1:
99s 27
8000
27
sys 10.9
by 0
955
20
6000
15000
4000
30008
feernenrrnnggeee steps
Meta stepp
Figure 8: Inner Adaptation convergence for early training stage (left)) and close-to-converggnnce training stage (right) for quadrotor.
In Fig. 9, we also include heat maps that demonstrate the changes in queries and values due the fine-tuning phase for two arbitrary system configurations.
27

---

Published in Transactions on Machine Learning Research (02/2026)
System
System B
-14
44
-14
(a) Adaption of queries and values during fine-tuning
(b) Adaption of queries and values during fine-tuning
Figure 9: Adaption of queries and values during fine-tuning for two arbitrary systems.
E
Context-Window Study
The context-windows choice application and task dependent in general, since it comes as a trade-off between accuracy and long-term prediction. We provide Table 7 comprehensive study different context-window lengths. In all of our results and studies long-term prediction capabilities of MetaSym.
context-window of 30 time-steps, because we prioritize the
Table 7: Mean Squared Error (MSE) with standard deviation (a) for different context window sizes. As expected for Markovian dynamics, the compounds the context-windov increases, however MctaSym retains good trade-off between long-ten term prediction and accuracy.
Context Window
MSE (#o)
2
0.2864 (0.2154)
10
0.4482 (0.3134)
20
0.9214 (0.8841)
30
1.5903 (1.6514)
50
3.5574 (3.8716)
100
8.4024 (7.7815)
F
Compute Resources
The dataset for the 10 x 10 spring mesh system and the associated baselines and benchmarks were Lambda cloud instance consisting of NVIDIA A100 with 40GB of GPU memory and 200GB of RAM. All other experiments were run on system containing 252GB of R.A,, 32 core processor and a 24GB NVIDIA RTX 4090.
G Hyperparameters for Benchmarks
This section includes the hyper-parameters of our framework, which we used to benchmark each system as outlined in Table 1. This ensures the reproducibility and completeness of our method.
28

---

Published in Transactions on Machine Learning Research (02/2026)
Table 8: Hyperparameter settings for MetaSym when benchmarking the spring-mesh. mesh.
Hyperparameter
Encoder
Decoder
Optimizer
Outer: Adamn
Outer: Adamw
Inner: Adam
Inner: AdamW
Learning Rate
Outer: 0.001
Outer: 0.007
Inner: 0.003
Inner: 0.01
Inner Steps
3
10
Context Window
N/A
30
Dropout
0.1
0.1
Layers
3
1
Attention Heads
N/A A
4
Early Stopping Epochs
110
312
Train / Val. dataset split [%]
80 / 20
80 /
20
Adapt / Infer. dataset split [%]
N/A
30 / 70
DrroCoonnctt
Table 9: Hyperparameter settings for MetaSym when benchmarking the open-quantum system.
Hyperparameter
Encoder
Decoder
Optimizer
Outer: AdamW
Outer: AdamW
Inner: Adam
Inner: AdamW
Learning Rate
Outer: 0.001
Outer: 0.008
Inner: 0.003
Inner: 0.03
Inner Steps
3
15
Context Window
N/A
30
Dropout
0.2
0.2
Layers
3
1
Attention Heads
N/A A
4
Early Stopping Epochs
294
354
Train Val. dataset split 1%)
/
/
Adapt / Infer. dataset split [%]
N/A
30 / 70
DropConnect
Table 10: Hyperparameter settings for MetaSym when benchmarking the quadrotor.
Hyperparameter
Encoder
Decoder
Optimizer
Outer: AdamW
Outer: AdamW
Inner: Adam
Inner: AdamW Outer: 0.01
Learning Rate
Outer: 0.001
Inner: 0.003
Inner: 0.008
Inner Steps
3
15
Context Window
N/A
10
Dropout
0.4
0.45
Layers
3
1
Attention Heads
N/A
4
Early Stopping Epochs
34
222
Train / Val. dataset split [%]
80 / 20
80 / 20
Adapt Infer. dataset split [%]
N/A
30
70
'rropComnett.
29


Reviews:
[{"review_id": "8PN96nUOiH", "reviewer": "Reviewer_7FVn", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes MetaSym, a two-part architecture for learning physical dynamics:\n\n(i) a SymplecticEncoder (a stack of SympNet blocks trained forward/backward to enforce time-reversal and preserve geometric invariants), \n\n(ii) an ActiveDecoder (autoregressive transformer with a meta-attention head; only query/value are adapted per system for few-shot transfer).\n\nThe system is evaluated on (i) a 400-D spring-mesh, (ii) an open quantum heterodyne-measurement task, and (iii) a quadrotor (sim→real fine-tuning). Table 1 (p. 9) reports lower MSE with fewer parameters than Transformers, DHNN/SymODEN, and FNO on these tasks; figures on pp. 7–9 illustrate long-horizon rollouts and real-world quadrotor predictions.  \n\nStrengths:\n\nBuilding on SympNets gives a clean way to encode canonical maps and time-reversibility; the bidirectional training is well motivated and clearly explained. \n\nThe datasets span classical, stochastic/open-quantum, and robotics domains, including real-world quadrotor adaptation with tight inference time.\n\nThe meta attention layer is simple and practically well motivated. \n\nWeaknesses:\n\nWhile the encoder is symplectic, the decoder perturbs the latent to model non-conservative effects. As written, it’s unclear how well the overall map preserves invariants over long horizons or what is guaranteed beyond “near-symplectic.”  Physics metrics (energy drift, momentum error, phase-space volume change) were not reported to fully understand the capability of the model.\n\nThe paper says ablations are in App. D, but the main text doesn’t isolate: (i) plain transformer decoder with vs without the SymplecticEncoder; (ii) symplectic pre-training vs random init; (iii) meta-attention (Q/V adaptation) vs full fine-tuning vs none. A small table/plot in the main paper would make the causal story crisp.\n\nThe quantum task is intrinsically noisy, however, any confidence interval (CI) plot is not provided.", "claims_and_evidence_yn": "No", "claims_explanation": "The paper is clearly written. The motivations and the claimed contributions are solid. There are some concerns regarding the symplectic structure claim, as the empirical results do not evaluate that. If the paper shows the evaluation, I would be happy to accept.", "audience_yn": "Yes", "audience_explanation": "It is useful for the community.", "requested_changes": "Please quantify with physics metrics (energy drift, momentum error, phase-space volume change) to understand whether the model is truly symplectic. Also, is there any theoretical guarantee on retaining symplectic structure?\n\nThe reviewer would like to see some ablation studies, as pointed out in the weakness. \n\n Please specify training split, adaptation shots/steps, units, and scaling of each channel, and whether the reported error is averaged across states or weighted. Also, add a runtime breakdown (encoder/decoder) to support the real-time claim, especially for the quadrotor experiment.\n\nWhat breaks if the context window is mis-specified (too short/long)?", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "ILiJ5rA7oM", "reviewer": "Reviewer_bTsD", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors present MetaSym, a deep learning framework that employs meta-learning in conjunction with a symplectic encoder and an autregressive decoder. The idea of the former is to preserve the Hamiltonian, thereby allowing for true time-reversible dynamics. The authors benchmark this on 3 real-world datasets, demonstrating few-shot adaptation and lower errors than larger SoTA models.", "claims_and_evidence_yn": "No", "claims_explanation": "I would say the results in this work support the claims, but that more evidence is needed. It isn't clear why the authors picked these particular benchmarks instead of somewhat obvious ones. For instance, examples from physics where analytical solutions are known would have been ideal, or at least cases where numerical solutions are available. When an architecture is presented and no connections are explicitly drawn between the architecture parameters and the generalization errors, very little can be gleaned from this work.", "audience_yn": "Yes", "audience_explanation": "I think this work is an important step in the right direction. The idea of combining meta-learning and symplecticity is quite timely, and has a wide variety of applications in classical and quantum mechanical systems, both static and dynamical.", "requested_changes": "I request that the authors add the following experiments/details, in my view crucial for acceptance of this work:\n\n1. A study of this approach on a baseline problem where an analytical solution is known, such as a harmonic oscillator or a more general N-body system such as the Calogero-Moser system.\n2. For each test problem, a clear description of why symplecticity is required, and possibly some measure of the efficacy of the architecture in maintaining symplectity. If the architecture analytically enforces it, why the errors on the order of 10^{-2}? How much should one expect?\n3. A clearer description of the symplectic architecture. What are alpha and beta? How are they chosen? What makes their particular network symplectic?\n4. Many of the experiment descriptions are confusing. For instance, to me, quadrature is a numerical integration rule. It seems to mean something else in this context, or I'm not following why that's a quantity of interest. The descriptions of all experiments are somewhat niche and not clear at all to a broader audience. The authors should update these and take care to define terms before they're used.\n5. The motivation for the chosen experiments isn't clear to someone like me who's not in this particular area. The authors could motivate things more clearly. If the experiments tell a story, what's the story? What other experiments might a broader audience who cares about Hamiltonians and symplecticity want to see?\n6. Appendix C talks about departure from symplecticity to allow modeling of dissipative systems. It would be great to see what happens on varied problems without any dissipation, where one expects strict symplecticity and time reversibility.", "broader_impact_concerns": "None.", "additional_comments": "I think the paper is reasonably well-written, with the exception of my concerns above.", "extra_scores": {}, "extra_text": {}}, {"review_id": "hVUmPSIGPv", "reviewer": "Reviewer_zQos", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "Summary:\nThe paper introduces MetaSym, a meta-learning architecture for quick adaptation of dynamical systems across different environments and parameter settings. The framework combines a symplectic encoder for preserving shared geometric structure with an autoregressive decoder that adapts to non-conservative dynamics through a meta-attention mechanism. The encoder is built from SympNet blocks and trained bidirectionally to encourage Hamiltonian consistency and stable long-term rollouts and the decoder incorporates per-system parameters and control inputs. The approach is evaluated on a spring–mesh system, an open quantum system, and a quadrotor, showing improved prediction compared to selected baselines.\n\nStrengths:\n1. Architecture choices tied to physically meaningful design: \nThe model’s components correspond directly to distinct physical roles for example 1) a symplectic encoder built from canonical flow blocks to capture shared conservative structure, 2) an attention-based decoder to flexibly model dissipation, control inputs, and measurement effects, 3) the bidirectional (Δt/–Δt) training of the encoder to reflect time-reversibility in Hamiltonian systems.\n2.  The method is tested on three very different domains: The paper evaluates the model on a mechanical system, an open quantum system, and a quadrotor sim-to-real task. This covers deterministic, stochastic, and controlled dynamics, which is an unusually broad range for a single framework. Showing good performance across all three suggests the approach is not tied to one specific type of system.\n3. Clear separation of shared vs. task-specific dynamics: The model assigns the shared, conservative structure of the dynamics to the symplectic encoder, whose parameters are meta-learned and kept global across all systems. In contrast, variations that differ from system to system are handled by a small set of task-specific query/value parameters in the decoder that are adapted in the inner loop.\n\nWeaknesses:\n1. Unclear to what extent generalization is possible: Because the encoder is frozen after meta-training, it cannot adjust to new systems whose conservative structure meaningfully differs (for example different stiffnesses, masses, or Hamiltonian parameters). This makes it difficult to assess how far the model can generalize beyond the training distribution and raises questions about the strength of the fast adaptation claim.\n2. The experiments do not show adaptation curves or comparisons with non-meta baselines under equal update budgets. In addition, the paper does not report any adaptation time, despite emphasizing “fast adaptation,” so it is unclear how efficient the inner-loop updates actually are in practice.\n3. Structure-preservation is claimed but not evaluated: The symplectic encoder is justified by its ability to preserve geometric structure, but no explicit metrics are reported. \n4.Limited baseline coverage: Although the paper discusses meta-learning methods such as iMODE in the related work, it does not include iMODE or similar task-adaptive baselines like CAVIA in the experimental comparison. These methods are designed for fast adaptation across dynamical systems and would provide a much clearer assessment of MetaSym’s meta-learning benefits.\n5. Limited literature review: The paper emphasizes fast adaptation and the preservation of physical structure as core motivations, but the related work section does not discuss several relevant meta-learning approaches in scientific machine learning. For example, there is a substantial line of work on meta-learning physics-informed neural networks (meta-PINNs) and parameter-aware operator-learning models that pursue similar goals of learning shared physical structure and adapting quickly to new parameters.", "claims_and_evidence_yn": "No", "claims_explanation": "Several of the claims are not fully backed by the evidence shown. In particular, the meta-learning / “fast adaptation” benefits are not isolated via adaptation curves, controlled comparisons to non-meta baselines, so it is hard to attribute the gains specifically to the meta-learning design. The structure-preservation motivation (symplecticity, time reversibility) is also not evaluated with explicit geometric or physical metrics, despite being central to the framing. Finally, the claim of being the first physics-based model to adapt across classical and non-unitary quantum systems is not adequately justified in comparison to the related work. Please refer to the weaknesses for more information.", "audience_yn": "Yes", "audience_explanation": "The paper fits within scientific machine learning and topics like physics-informed learning, structure-preserving modeling, meta-learning for dynamics which are of interest to many researchers in this area. Its attempt to use a single architecture across a broad range of classical and  quantum dynamics and would likely attract attention even if some claims need refinement.", "requested_changes": "1) Please clarify and discuss the extend of generalization. Please clarify what forms of system variation the method is expected to handle (e.g., changes in conservative structure vs. non-conservative effects). Since the encoder is frozen at test time, it would be helpful to explicitly discuss this limitation and provide experiments or analysis showing how the method behaves when conservative parameters (e.g., masses, stiffnesses) shift outside the meta-training distribution.\n2) Provide evidence isolating the meta-learning component. To support the “fast adaptation” claim, include adaptation curves or comparisons to non-meta baselines under equal update budgets.\n3) Overall the results section needs to be explained with more details. For example, since the symplectic encoder is motivated by geometric and physical invariance, please report metrics that directly support these claims.  \n4) The experiments would be stronger with comparisons to more directly relevant meta-learning approaches (e.g., iMODE or CAVIA)\n5) The related work section can be improved and expanded. For example by adding more directly relevant papers such as meta-learning PINNs. \n6) Please report adaptation time.", "broader_impact_concerns": "I do not see any ethical concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)