Physically Valid Biomolecular Interaction Modeling with Gauss-Seidel Projection
Authors: Siyuan Chen
Minghao Guo
Caoliwen Wang
Anka He Chen
Yikun Zhang
Jingjing Chai
Yin Yang
Wojciech Matusik
Peter Yichen Chen
Venue: ICLR 2026 Poster (2026)
Keywords: Biomolecular Interaction Modeling
Physical Validity
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: We introduce a differentiable Gauss-Seidel projection module that allows generative models to produce physically valid and highly accurate biomolecular structures in just two steps, achieving a 10x speedup over state-of-the-art baselines.

Abstract: Biomolecular interaction modeling has been substantially advanced by foundation models, yet they often produce all-atom structures that violate basic steric feasibility. We address this limitation by enforcing physical validity as a strict constraint during both training and inference with a unified module. At its core is a differentiable projection that maps the provisional atom coordinates from the diffusion model to the nearest physically valid configuration. This projection is achieved using a Gauss-Seidel scheme, which exploits the locality and sparsity of the constraints to ensure stable and fast convergence at scale. By implicit differentiation to obtain gradients, our module integrates seamlessly into existing frameworks for end-to-end finetuning. With our Gauss-Seidel projection module in place, two denoising steps are sufficient to produce biomolecular complexes that are both physically valid and structurally accurate. Across six benchmarks, our $2$-step model achieves the same structural accuracy as state-of-the-art $200$-step diffusion baselines, delivering ${\sim}10\times$ wall-clock speedups while guaranteeing physical validity. The code is available at https://github.com/chensiyuan030105/ProteinGS.git.

Published as a conference paper at ICLR 2026
PHYSICALLY VALID BIOMOLECULAR INTERACTION MODELING WITH GAUSS-SEIDER PROJECTION
Siyuan Chen
Minghao G.o
Caoliwen Wang'
Anka He Chen
Yikun Zhang'
Jingjing Chai
Yin Yang
Wojeiech Matusik2 Peter Yichen Chen
University of British Columbia
MITT CSAIL
SNVIDIA *Peking University
Foundry Biosciences
University of Utah
ABSTRACT
Biomolecular interaction modeling has been substantially advanced by foundation models, yet they often produce all-atom structures that violate basic steric feasibil- ity. We address this limitation by enforcing physical validity as strict constraint during both training and inference with a unified module. At its core is a differ- entiable projection that maps the provisional atom coordinates from the diffusion model to the nearest physically valid configuration. This projection achieved using Gauss- scheme, which exploits the locality and sparsity of the con- straints to ensure stable and fast convergence at scale. By implicit differentiation to obtain gradients, module integrates seamlessly into existing frameworks for end-to-end finetuning. With our -eeuss-Seidel projection module in place, two de- noising steps sufficient to produce biomolecular complexes that both phys- ically valid and structurally accurate. Across six benchmarks, our 2-step model achieves the same structural accuracy as state-of-the-art 200-step diffusion base- lines, delivering ~10x wall-clock speedups while guarantecing physical valid- ity. The code available atattps:: //tgt//gtttta com/ wwwp:hhuaannnnnn..... Protei nGS git
INTRODUCTION
End-to-end, all-atom protein structure predictors that integrate deep learning with generative model- ing are emerging as transformative tools for biomolecular interaction modeling (Senior et al., 2020bb Jumper et al., 2021; Abramson et al. 2024] Corso et al., 2023; Watson et al., 2023). These systems achieve unprecedented accuracy in predicting arbitrary biomolecular complexes and are shaping the future of computational biology and drug discovery.
Contrary to their high structural accuracy, current predictors often fail to satisfy a conceptually sim- pler but basic requirement: the physical validity of the all-atom output. This failure, also commonly termed hallucination, appears as steric clashes, distorted covalent geometry, and stereochemical errors (Fig. TT}. In practice, however, physical validity is a prerequisite: Atom-level physics vi- olations hinder expert assessment (Senior et al. 2020a), undermine structure- reasoning and experimental planning (Lyu al. (009)), and also destabilize downstream computational analyses such as molecular dynamics (Hollingsworth & Dror 2018 Lindorff Larsen et al. 2011).
The root cause of this problem lies the design of current predictors: as generative models, they are trained to match the empirical distribution of known structures, without enforcing physical validity as strict constraint in the training objective. Consequently, these models can assign non-zero prob- ability to non- conn--hsiccll configurations, flaw that persists even with large-scale training, both in data and model size, as observed in recent works (Wohlwend et al. 2024 Passaro al. 2025 Butten- schoen et al.) 2024 Team et al. 2025). Methods such Boltz- -Steering (Wohlwend et al. 2024) reintroduce physics as inference- time guidance by biasing the sampling process towards physically valid regions. Such guidance can reduce violations, but still cannot guarantee validity: with finite guidance strength and update steps, invalid configurations remain reachable.
"Equal contribution. Corresponning author.

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Published as a conference paper at ICLR 2026
Boltz-1
Boltz: Boltz-I-Stering 1-Steering
Protenix-ini
Ours
Ground Inuth
LDDE: 08177
LDDE 0.300 200 steps
LDDE: 0775 5 Steps
LDDE: 0832 2
200
Figure 1: Comparison on PDB 8B3E (protcin-ligane complex): global structure (top) and two zoomed views of the binding pocket (bottom). Protenix-Mini with 5 denoising steps exhibits backbone-ligand clashing. With 200 steps, Boltz-1 resolves the backbone but leaves clashes be- tween the side chain and the ligand. Boltz-1-Stend 1-teering removes clashes by using physics-informed potentials, but at the cost of large sampling steps. Ours yields physically valid results with only 2 denoising steps.
To close this gap, we elevate physical validity to a first-class constraint and enforce in both train- ing and inference by explicitly handling validity alongside generation: The denoising network first outputs provisional atom coordinates; a separate projection module then maps them onto the physi- cally valid set. Losses computed on the projected coordinates with gradients propagated through the projection module. This design allows the denoising network to redirect its capacity toward improving structural accuracy, offloading the task of avoiding physics violations to the projection module. This shift further removes the need for large denoising steps: at inference, sampling with few steps sufficient attain high structural accuracy with guaranteed physical validity, As such, the reduction in denoising steps is not due to specialized few-step diffusion techniques: it follows directly from treating physical validity as a constraint throughout both training and inference.
Concretely, we instantiate this decoupling with differentiable Gauss-Seide projection placed after the denoiser. solves constrained optimization that projects all-atom coordinates to the nearest physically valid configuration. The constraints are tightly coupled through shared atoms and their number scales with the number of atoms, making the optimization problem large. Within the train- ing loop, the module must also converge in handful of iterations with low memory per forward and backward pass. These requirements make pure first-order gradient descent methods impracti- cal, which require tiny steps and many iterations to converge. We therefore adopt Gauss-Seide scheme (Sand 2003), which sweeps over all constraints for a few iterations, enforcing each locally by updating only the affected atoms. It leverages locality and sparsity of the constraints and thereby yields faster, more stable convergence than gradient descent. The projection module is differentiable via implicit differentiation. We integrate it as a drop-in layer within existing biomolecular interac- tion frameworks (e.g., Boltz (Wohlwend et al. (024 and finetune end-to-end. At inference time, 2 denoising steps suffice produce structurally accurate, physically valid protein complexes.
We evaluate our method on six protein-complex benchmarks: CASP15, Test (Wohlwend et al. 2024]. PoseBusters (Buttenschoen et al., 2024), AF3-AB (Abramson al. 2024). dsDNA, and RNA-Protein (Ma et al.] 2025). Comparisons include 200-step generative models (Boltz-1, 1, Boltz- 1- Steering (Wohlwend et al. (024), Boltz-2 (Passaro et al. 2025), and Protenix (Team et al./ 2025) and the few-step baseline Protenix-Mini (Gong et al., 2025). Despite using only two denoising steps, our model achieves competitive structural accuracy while guaranteeing physical validity. For runtime, model delivers ~100 wall-clock speedup
baselines. Our study largely closes the gap between guaranteed physical validity and state-of-the-an structural accuracy under few-step sampling, enabling 2-step all-atom predictions with an order-of- of-magntuude faster inference.
2 RELATED WORK
Protein Structure Prediction. Deep learning- based foundation models are reshaping the task of protein structure prediction that produces 3D coordinates from input sequences (Senior et al. 2020b Jumper et al.) 2021). End-to-end variants extended this paradigm to complexes and large-scale training and inference (Back et al. 2021 Evans et al., (022). AlphaFold-3 further moved to all- atom biomolecular interaction modeling with a diffusion sampler (Abramson et al. 2024), followed by improvements on generative models, such as Boltz-2 (Passaro et al.] 2025) and Protenix (Team) et al. 2025). These methods typically require hundreds of denoising steps. Few-step variants such
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as Protenix-Min (Gong al. 2025) reduce sampling to two steps while maintaining accuracy, yet still lack guarantees on physical validity. Boltz- - (Wohlwend et al.) (024) is the first to explicitly target this issue by steering sampling with physics-nffommed potentials, but the guidance is soft and cannot preclude violations. By contrast, we introduce a Gauss-Seidel projection layer that enforces physical validity for both training and inference.
Diffusion Models with Tilted Distribution Sampling, Diffusion models have demonstrated strong capabilities in producing samples given input conditions (Ho et al. 2020 Dhariwal & Nichol 2021
Song et al. (02T). However, in many cases, users require the generated samples to satisfy cer- tain specific constraints. There are primarily two types of constraint-aware diffusion methods: (i) Feynman-K Kac (FK) steering- baared conditional sampling methods (Trippe et al.) [023] Singhal et al.) 2025), which steer the sampling toward regions that meet desired conditions through reweighting of the path distribution in the diffusion process, and (ii) manifold-based diffusion models (Bortoli et al. 2022 Elhha et al., 2024), which explicitly construct constrained manifold, ensuring that gen- erated samples inherently satisfy the desired constraints. However, FK-steering typically requires large number of sampling steps, and explicit manifold construction can be computationally expen- sive when constraints complex. We introduce Gauss-Seide projection module that enforces constraints in training and inference. thereby circumventing both excessive sampling iterations and the need for expensive manifold construction.
Constraint Enforcement via Position-Basee Dynamies. Enforcing physical constraints has long been central physics simulation across computational physics and computer graphics. To handle boundary conditions (Macklin et al. 20144), collisions, contact, friction (Bridson et al.] (0022), and bond stretching (Ryckaert et al., 1977). many methods have been developed to keep simulations physically consistent. Among these, position-based dynamics and its extensions (Mülee et al. 2007
Macklin & Muller 2016, Chen et al. (024) iteratively project particle positions to satisfy predefindd
constraints, proving effective for soft bodies, fluids, and coupled phenomena (Bender et al.) 2014).
By updating positions directly instead of integrating stiff forces, these methods remain stable under large timesteps and avoid costly global solves. Since constraints are enforced through small, local projections, the overall complexity scales nearly linearly with the number of constraints, which aligns with our goal of generating proteins that respect local physical constraints.
Finetuning Difftusion Models. Finetuning has become essential for tailoring diffusion models to specific downstream tasks. To enhance controllability, many recent works employ reinforcement learning (RL) strategies: (i) methods such as DRaFT (Clark et al.] (024) and DPPO (Ren et al.] 2024 treat the denoising trajectory as policy directly optimize reward functions; (ii) approaches like DPOK (Fan et al.] 2023) and DiffPPO (Xiao et al. 2024) leverage preference data or human feedback. While effective at steering generation, these RL-baaed methods often incur high compu- tational costs or exhibit sensitivity to the quality of the feedback signal. In contrast, we introduce Gauss-Seidel projection during training to finetune the model via strictly defined physical con- straints, circumventing the instability of policy gradient estimation.
PRELIMINARY
All-Atom Diffusion Model. Proteins can be represented at all-atom resolution (Abramson et al., 2024 Wohlwend et al. 2024 Passaro et al. 2025), where a structure is specified by the Cartesian coordinates of all atoms & E N with N the total number of atoms. This representation en- ables direct modeling of side-chain orientations, local packing, and explicit inter-atomic geometry. AlphaFold3 (Abramson et al.) 2024) introduced an atom-level diffusion approach that iteratively denoises atomic coordinates from Gaussian noise. Conditioned on features produced by the atom- attention encoder, MSA module, and PairFormer, a denoising network runs for hundreds of steps (typically 200) to generate the all-atom prediction &
Physics-Guided Steering. All-atom diffusion sampling is stochastic and therefore cannot guaran- tee physical validity. To address this, -oSte-i-S (Wohlwend et al., 2024), built upon the Feynman-Kee Kac framework (Singhal et al.) 2025), employs physics-informed potentials. At each denoising step, these potentials score the validity and tilt the learned denoising model toward a lower-enese eever-enegyy configuration. The energy is weighted sum of seven terms: tetrahedral atom chiral- ity, bond stereochemistry, planarity of double bonds, internal ligand distance bounds, sterie clashes avoidance, non-overlapping symmetric chains, and the preservation of covalently bonded chains.
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Sequence
Atom Attention Encuder
MSA Module
PairFormer Module
random
Denoising Model
x2
Gauss-Seldel Projection Module
Physical Validity Constraints
Diffusion Loss
...
C(x)
Xyco
Figure 2: Enforcing physical validity during both training and inference via our Gauss-Seidel projection module. Provisional all-atom coordinates from the diffusion model are corrected by Gauss-Seidel projection that sequentially resolves local constraints, each acting atoms and updating coordinates in place. The module is differentiable via implicit differentiation, allowing seamless integration into training. The same projection is applied at inference, ensuring physical validity and enabling accurate predictions with as few as two denoising steps.
small set of
Since this tilted distribution cannot be sampled directly, the method uses importance sampling to draw multiple candidates, which are then refined with gradient descent updates further reduce violations. Although this steering reduces physical errors, it remains soft approach: it biases the sampling process toward validity during inference rather than embedding validity into the training.
4 APPROACH
Our goal is to predict all-atom biomolecular complexes that are both structurally accurate and physi- cally valid. Given provisional coordinates produced by diffusion module, we enforce validity with a Gauss-Seide projection module that sequentially resolves physical constraints on the affected atoms (Sec. 4.1 and 4.2) The module differentiable, allowing training losses be computed on the projected atom coordinates, and gradients to propagate back through the projection module via implicit differentiation (Sec. 4.3) Crucially, this module enables accurate inference with only two denoising steps while guaranteeing physical validity. Fig. iillustrates the overall pipeline.
4.1 PENALTY-BASED FORMULATTION FOR PHYSICAL VALIDITY
Given the Cartesian coordinates all constituent atoms x produced by the diffusion module, our projection module finds the nearest physically valid coordinate configuration:
Xproj arg min x11 s.t. C(x)= 0,
(1)
where C(.): Nx3 + TM concatenates all constraints that characterize physical validity, and its
zero set defines the feasible space. We follow Boltz-1-Steering (Wohlwend et al. 2024) to define the constraints. Detailed formulation is provided in Appendix [ Since all terms locally defined on pairs or four-atom groups, the total number of constraints m scales with atoms and can be large. For instance, for PDB 8TGH which consists of 470 residues with 7, 082 atoms, is 13M.
standard approach to solving the constrained problem Eq. D is the penalty method (Boyd & Van- denberghe 2004), which the constraints are incorporated an exterior penalty into the objective:
Xproj = arg min E(x) + +||x -
x1))).
T E(x) = T -C((x,,,
(2)
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where choose quadratic penalty E-)) and a Rm is block-diagonal matrix of penalty coefficients. For sufficiently small a (i.e., large weights a )) Eq. 2 as a numerical realization of the card-constrained projection: the minimizer satisfies the constraint up to numerical tolerance. In our implementation, we set j = 10- or 10-6 to enforce a tight feasibility threshold.
The penalty formulation defines deterministic mapping & - Xproj that links the provisional coor- dinates of the diffusion module to a physically valid output. Solved on its own, this mapping already functions plug-and-play post-processing module for biomolecular modeling frameworks. Since the projection can resolve invalidity efficiently, we aim to explicitly decouple validity enforcement from the network training by integrating the module within the training loop. Concretely, we insert the module immediately after the diffusion module and compute training losses on the projected coordinates. the following sections, we detail the forward solver (Sec. 4.2) and the backward pass via implicit differentiation (Sec. 4.3 enabling the module to operate as a fully differentiable layer.
4.2 GAUSS-SEIDEL CONSTRAINT PROJECTION
Solving Eq. 2 alone is classical and admits many off-the-shelf solvers. Embedding it as a differ- entiable module within a biomolecular modeling pipeline, however, imposes stricter requirements: very fast convergence per forward pass and training step, and stable convergence under stiff, tightly coupled constraints. Pure first-order gradient descent, used in Boltz-1-Steering (Wohlwend al.) 2024], is ill-suited: it requires small step sizes and prohibitively many iterations to converge to a zero constraint penalty, making unrolled end-to-end training slow. We therefore propose Gauss- Seidel projection that exploits locality and sparsity by sequentially updating only the atoms affected by each constraint, thereby achieving fast and stable convergence.
We consider the first-order optimality condition for Eq. 2
VC(x) CC(x + (x
0.
We then introduce the Lagrange multiplier X(x) = - a -ae-aa-CCi CC(x) by following (Stuart & Humphries 1996) and (Servin et al.) (0066) to obtain the coupled system:
(x x) - vC(x) T A(x) = 0, C(x) - xx(x) = O.
(4)
This system is nonlinear; we solve it by iterative linearization about the current iterate (x("), )(1)). At iteration n, the linearized system is:
I
(
Ax
((x( a xm)
(C(((())
AA
with updates x 1n+1) (n) + Ax and (m+1)_ (m) + AA. can be shown that with initialization
(0) x and (0) O, the linearized iterates converge to solution of the original nonlinear system
Eq. 4 See Appendix [ for a proof. Applying the Schur complement (Zhang 2005) with respect to I
gives the reduced system for the multiplier update:
(VC(x!") (m) (C)(()))) +a)AA- = - - ax
(5)
and the position update Ax- = VC(x("))TAA.
The primary computation in each iteration is solving the large linear system Eq. Rather than constructing the full linear system, which is memory-intensive and costly, we employ a Gauss- Seidel scheme that iteratively sweeps over the constraints: During each sweep, the method addresses one constraint at a time by updating only the atom coordinates it affects. These new coordinates are used immediately when processing the next constraint. After small number of sweeps, the system converges to configuration where all constraints are satisfied. This Gauss-Seidel scheme significantly more efficient than a global solve because it effectively leverages the locality and sparsity inherent in Eq. 5 Specifically, for the j-th constraint, the Gauss-Seide update is
(x(x() - a,2(m) (n)
Ad,- VC, +aj
j = 1, m.
(6)
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Algorithm 1 Forward Solver of Gauss-Seie Seidel Projection Module.
Algorithm 2 Backward Pass of Gauss Seidel Projection Module.
Input: Atom coordinates &; constraints and their gradients {Cj. VC, در=ر رリ penalty co-
Input: Projected coords Xprofi upstream gra- dient 0L/OXprujj {C,, VCj, V²C, "
efficients {ax د sweeps T
{or { tolerance 8: iters K
Output: Projected coordinates Xproj
Output: Gradient w.r.t. i.e. 0T./0%
1: x(0) 6- 8; ( -- 0
1: + (aL/0xpro)
2: for n 1 T do
sweeps
2: Compute A (xproii) 1 as in Eq.
3: for jo 1 to m do
3: for k = 1 to K do
4:
Compute A); using Eq. 6
4: CG update step for A @xpius)
T
5: x(a+1) t (() + VC, (x()))A
6: A (m+1)
(m)
+ AAj
5:
if 12 -R-1ll S E then
6:
break
7: end for
7: end
8: end for
8:
end for
9: return Xproj f- (T)
9: return
0L z
Processing all constraints sequentially and repeating for small number of sweeps (20 in our imple- mentation) yields fast and stable convergence in practice. To maximize computational throughput, implement the solver GPUs (details in Sec. 5.1. Algorithm Joutlines the forward process of the Gauss-Seidel projection module.
Remark 4.1 (Fast Convergence and Constraint Satisfaction of Gauss-Seidel Projection) The convergence of the Ccuus-Seider projection of quasi-second order. Each update yields a monotone decrease with O(1) work per constraint (hence (m)) per sweep). With strict tolerance and sufficiently small a, the final iterate satisfies the first-order optimality condition and realizes the hard-constrained projection. See Appendix foor details.
4.3 DERIVATIVES VIA IMPLICIT DIFFERENTIATION
In the forward pass, the module projects x into Xproj Xproj (x), which is used to calculate the training loss L [xxproj)- The corresponding backward pass is to compute the gradient of this loss with respect to the input of the projection module, ux
Making the projection module differen- tiable the key to enabling few-step sam- pling- A standard diffusion module is trained handle two tasks simultane- ously: achieving structural accuracy and ensuring physical validity. relies on a large number of denoising steps its ef- fective capacity to perform both. Con- sequently, when the step budget is small, the model's structural accuracy degrades significantly, shown in Fig. finetuning the network with our differen- tiable projection module, we enable a de- coupling of responsibilities. The denois- ing network learns focus exclusively on recovering structural accuracy, offloading the task of ensuring physical validity to the projection module. This decoupling makes highly accurate predictions possi- ble even with very small step budget.
By
Gauss-Seidel Projection w/o Finetuning
Gauss-Seidel Projection w/ Finetuning
Figure 3: Importance of differentiable projection and finetuning. A post-hoc projection without fine- tuning with a 2-step sampling, while ensuring physical validity. fails to recover the a-helical secondary struc- ture (left). Integrating the projection as differentiable layer and finetuning diffusion module restores the helix and improves structural accuracy (right).
Backpropagating through the Fruss-Seided projection is non- con-trivial because it is implemented with an iterative solver. A naive approach, such unrolling the forward iterations for automatic differen-

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tiation, is prohibitively memory iimooryinttessvee and thus impractical for end-to-end training. We resort to implicit differentiation, a technique from sensitivity analysis (Burezyńíski et al. 1997). Specifically, differentiate the first-order optimality condition in Eq. with respect x:
Oxproj
(H(Xproj) +1)
I,
H(x) =
(x) [x) (x) (C,(x)] (7)
0%
which is evaluated at x Xproj.
By the chain rule
aL proj
0x 8%
we obtain the adjoint system
8%
ÖT. HH (Xpoo) + II" z ( (xxproo ) and
θT. 0x
T.
Analogously to the forward pass, the backward pass requires solving an additional linear system. We use conjugate gradients (CG) to solve Eq. [8 whose convergence is guaranteed near feasibility (see Appendix K for details). For runtime efficiency, we implement the CG solver on the GPU. Algorithm outtines the backward process of the projection module.
EVALUATION
In this section, evaluate our method's structural accuracy and physical validity through series of experiments, We report quantitative results on six evaluation metries (Sec. 5.2).. We also provide qualitative visualizations that compare protein struc- tures and demonstrate the physical validity (Sec. 5.3 Furthermore, we analyze the convergence speed and wall-clock runtime of our approach (Sec. 5.4). Finally, we conduct an ablation study to
benchmarks using standard
isolate the impact of our Gauss- Seidel projection (Sec. 5.5
5.1 IMPLEMENTATION DETAILS
With the differentiable Gauss-Seidel projection module in place, we finetune pre-trained all-atom diffusion network specifically for 2-step sampling. During finetuning, we retain the original noise parameterization and denoising objective. At inference time, use the inference-time sampling schedule of Protenix-Mini (Team et al.) 2025). A Gauss-Seidel projection is applied to the model's output to guarantee physically valid final structure. Appendix H provides pseudo-code and com- plete algorithms for both training and inference with our projection module. We implement the differentiable Gauss-Seide projection the GPU using the WARP library (Macklin et al.) 2024). In the forward pass, constraints are partitioned into batches with no shared atoms, allowing them to be processed in parallel. Coordinate updates accumulated with atomic operations. We use 7=20 sweeps. The backward pass solves the adjoint linear system with a conjugate gradient solver fully on GPU, parallelizing per-constraint operations with atomic accumulation. We the tolerance to 1=10- and the maximal number of iterations to 1000. The projection module integrates with PyTorch as custom autograd layer, avoiding the overhead costs of host-device copies and context switching. For training, we finetune Boltz-1 [Wohlwend et al.) (2024) from open-source pretrained weights on the RCSB dataset (Burley et al. (022) (~~ 180,000 structures) using 8 NVIDIA H100 GPUs over two weeks. All evaluations are conducted on NVIDIA A6000 GPUs. For baselines and evaluation protocol, see Appendix M for details.
5.2 QUANTITATIVE RESULTS
Table [ andd summarize the comparison across six protein-complex benchmarks. Among meth- ods, ours uses the fewest denoising steps (2 steps). It achieves state-of-the-arr accuracy in the few- step regime: Compared to Protenix-Mini (5 steps), our model improves complex- and interface- level LDDT across most benchmarks. Compared to 200-step baselines, the accuracy of our method is competitive. On Test and PoseBusters, our performance comparable to Boltz-1 1 and Protenix across multiple metrics while approaching Boltz-2, which requires 100 times more sampling.
For physical validity, our approach unequivocally best: it achieves 100% validity across all bench- marks. Methods that do not enforce validity (Boltz-1, Protenix, Protenix-Min Minii) show frequent

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Table 1: Quantitative results on datasets (Part 1). Our method uses only 2 denoising steps yet always guarantees physical validity and achieves competitive structural accuracy compared to baselines. Dark green cells denote the best results among the few-step methods, while light green cells denote the best results among the st0-step methods. We also include the results of Al- phaFold33 (Abramson et al. 2024) for references.
Metric
# Denoise Steps
Complex LDDT
Prot-Prot ILDDT
Pio-Prot ILDDT
DockQ > 0.23
I.RMMD < 2
Physical Validity
TM-seore
Method
AlphaFold3
200
0.05
0.05
0.58
0.67.1..33
0.36.10.28
0.39.00.08
0.7410..11
Boltz-1
200
0.64=0.06
0.53:0.16
0.58
0.73
0.20:1::20
0.37
0.65
CASPIS
Protenix
200
0.65
0.7510.17
0.391:038
0.561:.33
0.28:1:025
0.38:10.08
0,65
Boliz. I Steering
200
0.64:10.06
0.45110.11
0.58
0.71
0.15:10.15
0.37.10.07
1.00..0.00
Boltz-2
200
0.67=2006
0.74:1.112
0.71
0.73
0.42:10227
0.40:10.08
1.00
Protenix- Mini
10
0.60=0006
0.54
0.33
0.38::0,26
0.37:10.27
0.35
0.66::0.13
Protenix Mini
5
0.59
0.56
0.491.0.37
0.41
0.37 0.224
0.361.0.07
0.64
Ours
2
0.63=0006
0.41
0,32
0.71
0,36:1:.28
0.36::0.07
1.00
AlphaFold3
200
0.82=0.01
0.50+::.33
0.58=0.07
0.72+:0.05
0.57::0.06
0.85:20.22
0.62+00.22
Boltz-1
200
0.80=0.11
0.44
0.52=0.07
0.69=
0.56+00.06
0.83=0002
0.44
Test
Protenix
200
0.80-0.01
0.44
0.56
0.68
0.55:10.06
0.83:10.02
0.51 0.504
Boltz-I-Steering 1-Steering
200
0.80=0.01
0.44
0.55
0.68-+0.06
0.55:00.06
0.83:20.02
1.00:10.00
Boltz-2
200
0.83=0.01
0.52
0.62=0.08
0.74#:005
0.60+ 06
0.86+0002
1.00=00.00
Protenix Mini
10
0.73-20022
0.37-1:0.3
0.57
0.55:10.06
0.51:100005
0.76:10.02
0.45
Protenix-Mini
5
0.73=0.02
0.37+:0.03
0.50
0.55+40.05
0.47:0006
0.76+:0.02
0.5:00.04
Ours
2
0.79-0.01
0.40..0..3
0.52
0..06
0.52.10.06
0.80
1.00
AlphaFold3
200
0.93=0.01
0.57+-0.06
0.78=0.02
0.65:+0.08
0.73+t0.04
0.90+00.02
0.48
PoweBusters
Boltz-1
200
0.92=0.01
0.50:1:006
0.69::002
0.59:10.09
0.62:10.05
0.02
0.28:10005
Protenix
200
0.92=0001
0.53
0.73+0002
0.61
0.67:00.4
0.89+20.2
0.49:20.05
Boltz-1-S 1-Stecring
200
0.92-0001
0.52+00.6
0.69=0.02
0.59:
0.62:00.05
0.88+0.04
0.95=00.03
Boltz-2
200
0.94=0.01
0.57::0.06
0.02
0.63:::009
0.79:10.04
0.90::0.02
0.88:10.04
Protenix-Mini
10
0.90=0.01
0.49::0006
0.72-0022
0.57:+0.09
0.67::0.04
0.87+00.22
0.37=20.05
Protenix Mini
5
0.90.10.01
0.49110.06
0.7010.02
0.089
0.63
0.871.0.03
0.27
2
0.9100011
0.50:10:06
0.69:1002
0.59
0.61:1:.05
0.88
1.00::0.00
clashes and physical errors, whereas steering-based methods (Boltz-1-Steering Boltz-2) improve but still admit failures because validity is used as guidance rather than a constraint. These non- physical outcomes limit downstream usability, whereas our predictions satisfy validity and preserve high structural accuracy with only two denoising steps.
5.3 QUALITATIVE RESULTS
Fig. and (in Appendix present qualitative comparisons, showing that our approach consis- tently yields physically valid structures while preserving high accuracy. For PDB 7Y9A (with an Saceetl---glucosamee
ligand), which is used to assess ligand pronteinroein interface quality, all base- lines produce severe ligand-protein atom clashes, whereas our method eliminates these violations. For PDB 7XYO (557 residues with extensive secondary structure), which is used to evaluate large, structured assemblies, Boltz- Boltz-I-Steering and Protenix show inter-chain collisions. Protenix- Mini avoids clashes but introduces large distortion, particularly in the second coiled segment. In contrast, our method produces clash-free structures while maintaining structural accuracy. Addi- tional examples are provided in Appendix A
5.4 ANALYSIS
Analysis on Convergence Speed. We compare the convergence speed of our Gauss-Seidel pro- jection against gradient descent guidance (Boltz-I-Steering). We add Gaussian noise to the ground truth coordinates of PDB 8X51 with four levels (o E {160, 120, 80, 40)) and run the same diffu- sion module, with either gradient descent guidance or Gauss-Seidel projection to enforce physical validity. We visualize the potential energy versus the number of update iterations in Fig. [5] (left). Gauss-Seidel projection reduces the physics potential rapidly and converges within 20 iterations for all noise levels. In contrast, gradient descent exhibits oscillations and long tails, requiring far
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Boltz-1 PDB 7Y9A
Boltz-1-Sterrngg
Boltz-2
Protenix
Protenix-Mini
Ours
POB TXYO
Figure 4: Qualitative comparison with baseline methods. The red color highlights physically invalid predictions, such as atomic clashes. Our approach consistently guarantees physical validity. The first example demonstrates a ligand-protein steric clash, where the ligand atoms physically penetrate the receptor surface. This corresponds to constraint 1 (Steric Clash) in Fig. [1 The second example demonstrates an inter-chain overlap, where atoms from Chain A and Chain occupy the same space. This corresponds to constraint (Overlapping Chains) in Fig. 11
more iterations to achieve a comparable reduction. This gap reflects the algorithms: Gauss-Seided projection is of quasi-secone order and thus converges faster than first-order gradient descent.
we measured the wall-clock infer- ence time (including diffusion and physical validity enforcement) on CASP15. The results, binned by atom count, are shown in F Our 2-step predictor with Gauss-Seida projection is con- sistently the fastest. It achieves median speedup of ~9.4 ~9.5 over Protenix. Compared with Protenix-No Mini with 5 denoising steps, our method achieves ~2.3 speedup. Compared with Boltz-1 1-Sterinn, the speedup is 23-46x. The gains stem from the reduction denoising steps and our GPU implementation of the projection module.
Wall-clock Runtime Comparison. assess practical speed,
the 200-step Boltz-1 baseline and
5.5 ABLATION STUDY
To isolate the impact of our differentiable Gauss Gauss-Seide Seidel (GS) projection, conduct ablation study on CASP15 using the 200-step Boltz-1 model as a reference. We compare four 2-step sam- pling variants, presented in the order reported in the inset table: (i) a vanilla 2-step sampler with no mechanism for physical validity: (fi) adding gradient descent (GD) guidance at inference; (iii) adding GS projection inference (only as post-processing) without fine-tuning: (iv) finetuning with GS projection and also applying it at inference (our full method).
Moving from the 200-step baseline a 2-step sampler without guidance re- duces both accuracy and physical va- lidity. Adding gradient descent guid- ance at inference improves validity but fails to recover the structural ac- curacy. Using Gauss-Seidel only as
Complex Physical LDDT Validity
200-step
0.6406
0.6345
2-step (no guidance, no finetune)
0.5920
0.6327
+ GD at inference, no finetune
0.5942
0.7345
GS at inference, no finetune
0.5889
1.00
post-processing guarantees validity, but at the cost of slightly degrading the accuracy. Our full setting closes the gap to the 200-step baseline with two denoising steps and physically valid. These results show that while Gauss-Seidel projection necessary to enforce validity, making it differentiable is critical for high structural accuracy in the few-step regime.
GS at inference, finetune 0.6321
1.00

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150
Bolte-1
Protenis-Mial
Frortal Larro
Wall--lock Running Time (s)
Bolte Steering
Ourr
100
Prodeols
250
200
Goodica
Porcate Exerp
200
50
Number of Atoms
Figure 5: Convergence and runtime. Left: Potential energy vs. iteration PDB 8X51 under four noise levels (o (160, 120, 80, 40 Gauss-Seidel projection (orange) converges to tolerance within 20 iterations; gradient descent guidance (blue) oscillates and decays slowly. Right: Wall- clock inference time by atom-count bin on CASPI5. Ours is ~10x faster than 200-step baselines and ~2.3x faster than Protenix-Mini, while maintaining validity. Runtime includes diffusion and validity module; measured on the same hardware.
Table 2: Quantitative results on datasets (Part M.. Dark green cells denote the best results among the few-step methods. Light green cells denote the best results among the 200-step methods.
Metric
Demoise Steps
Complex LDDT
Prui-Prot iLDDT
DNA-Prot
RNA-Prot ILDD
Physical Validity
TM-score
Method
Roltz-1
200
0.80+0.02
0.12=0005
0.61
0.00=0.00
Protenix
200
0.80=0.03
0,23=0.06
0.6420.06
0.01=0.01
AF3-B
Boltz-1-Stecring
200
0.7920.03
0.13=0.06
0.62+0.07
0.8920.11
Boltx-2
200
0.8740.03
0.55=0.10
0.7240.07
0.89+0.1 11
Protenix-Mini
10
0.77=0.03
0=0.40
0.60e0.06
0.00=0.00
Protenix. Mini
0.77=0.03
0:1044
0.59:0.06
0.00=0.00
Ours
0.78+0.02
0.14=0.05
0.60=0.06
1.0+00.00
Boltz-1
200
0.8540.04
0.7640.07
0.8920.06
0.03+0.03
Protenix
200
0.86#0.04
0.7840008 08
0.8920.06
0.04=0.03
dsDNA
Boltz-I-Steering
200
0.8540.04
0.7740.07
0.9020.00
L.000.00
Boltz-2
200
0.94g0.02
0.9140.05
0.93g0.06
1.0000.00
Protenix-Mini
0.8240.05
0.71+0.09
0.85+0.07
0.18=0.10
Protenix-Mini
0.8110.05
0.7140.09
0.8520.07
0.2420.09
Ours
0.80+0.04
0.71+0.07
0.88=0.00
1:000.000
Boliz-1
200
0.74x0.04
0.30
0.85e0.11
0.00=0.00
Procein
Protenix
200
0.75+0.05
0.34#0.09
0.82+0.11 12
0.02+0.02
Boltz-1 1----115800 --ttering
200
0.7340.05
0.30g0.07
0.83g0.11
0.9620.04
Boltz-2
200
0.8860.05
0.73
0.88+0.11
0.95
Protenix-Miis
0.7340005
0.3820.10
0.8020.13
0.02+0.02
Protenix Mini
0.7240.05
0.3620.10
0.79
0.05
Ours
0.7240.05
0.32+0.06
0.82e0.11
1.000.000
6 CONCLUSION
In this work, we propose differentiable Gauss-Seide projection module that enforces physical validity during both training and inference for all-atom biomolecular interaction modeling. Framed as constrained optimization, the module projects the coordinates produced by the diffusion module to configurations without violating physical validity. Finetuning the diffusion module with this projection achieves physically valid and structurally accurate outputs with only 2-step denoising, delivering ~ 10x faster inference than baselines while maintaining competitive accuracy.
Limitations and Future Work. While our projection module substantially reduces the number of denoising steps, single-step inference remains out of reach. Future work will focus on achieving it by combining our projection module with one-step diffusion training techniques.
Reproducibility Statement. To ensure the reproducibility of our work, we provide detailed ex- perimental settings in Sec. 5.1 the pseudocode for the Forward Solver and Backward Pass of the
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Gauss-Seidel projection module in Sec. 4.2 and Sec. 4.3 the pseudocode for training and inference algorithms in Appendix H the details on the baselines and the evaluation protocol in Appendix M and the complete formulations of the physical validity constraints in Appendix L
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10
3.0
Porward
107
Potential Emerryy
101
10-17
Memory Cost (GB)
1.mm
2.0
8.3
Time
1.0
10"9
10111
20
Iterations
09 09 100 2316 164
Number of Tokens
512
640
Figure 6: Convergence and computational cost analysis of the Gauss-Seidel projection module. Left: Potential energy plotted over iterations for a values ranging from 10 to 10 10- Large values (>10-5) result in persistent oscillations without full convergence. An intermediate value (0=10-5) exhibits transient oscillations before reaching zero potential, while smaller values (10-6 10-3 en- sure rapid and stable convergence, indicating robust constraint satisfaction. Right: Computational benchmarks showing forward/backward time costs and additional memory overhead across varying token counts (256, 384, 512, and 640).
A ADDITIONAL QUALITATIVE RESULTS
Fig. showws more qualitative comparisons with baselines. We also visualize full protein structures predicted by our method and compare them with the ground truth from six datasets, shown in Fig. [0] The results demonstrate that our model accurately predicts physically valid structures.
B UNIQUENESS OF THE GAUSS- -SIDEL SOLUTION
The Gauss-Seidel scheme guarantees uniqueness for linearized subproblems. Theoretically, while ordering and initialization govern the convergence speed of the Gauss-Seide scheme, they do not influence the solution's 'ss uniqueness (Golub & Van Loan 2013).
We investigated this empirically on the PoseBusters dataset by performing five independent projec- tion runs for each denoised structure. The computed pairwise RMSD among outputs was 0.0000 + 0.0000 Å. This confirms that, in practice, the projection module behaves stable mapping, reliably converging to a numerically identical solution.
C PRACTICAL CHOICE OF AND ITS SENSITIVITY OF CONVERGENCE
To assess how the parameter influences the convergence of the Gauss-Seidel projection, we ducted an ablation study on the protein PDB 7WUY, evaluating both convergence behavior and predicted accuracy across six values (10 ..... 10-7)). As illustrated in Fig. 6 moni- tored the potential energy after each iteration and observed two distinct behaviors: For large physically valid configuration. While the potential
(10-2,10-3 10- fails converge energy decreases initially, it exhibits persistent oscillations rather than converging to zero. This in- dicates that the constraint penalties are not enforced strongly enough to resolve physical violations. For small (S 10-5), at 1=100- the energy oscillates transiently before eventually converging to zero, confirming that all constraints are resolved. For smaller values 1a=10-6, 10-7), the projection consistently and rapidly converges to valid state (zero potential energy), demonstrating fast and stable constraint satisfaction. These results confirm that selecting 100 critical for ensuring the projection reliably maps physically valid configuration.
D COMPARISONS WITH DIFFERENT DENOISING STEP BUDGETS
We conducted experiments comparing Boltz-1, Boltz-2, and our method (built on Boltz- 1) across denoising step budgets of 2, 5, 10, and 200 the CASP15 dataset. As shown in Fig. [ our method
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Comple L.IT
Physeea Valalry
3
Decobing Steps
Dessising Stepo
Figure 7: Comparison of Boltz-1, Boltz-2, and our method across 2-, 5-, 10-, and 200-step bud- gets. Our method consistently outperforms the Boltz-1 baseline while maintaining strong physical validity. Although Boltz-2 achieves higher Complex-LDDT scores, integrating module into the Boltz-2 preserves this accuracy while significantly boosting physical validity at low step budgets.
PDB TNCC, RMED 02400
PLL RMSD 0151
PDB 7ELT.
RMSO: 11111
POBBAEM, RMBID (1778
Hefore Caaze-Siiil Projection
After GazweSeidd Projection
Before Canse--eidid Projection
After Canve-Seided Projection
Figure 8: Visualization of structural corrections by Guass-Seidel projection. The projection effectively resolves steric violations and restores physically valid geometries. We showcase four representative cases from the PoseBusters dataset. PDB 7P4C: A distorted a-helix clashing with neighboring strand is corrected. PDB 7PL1: A self-entangled coil is disentangled, resolving clashes. PDB 7ELT: Ligand-sidechain interactions and invalid internal ligand geometries are resolved. PDB 8AEM: Ligand-related collisions are removed. For each case, RMSD values were computed to quantify the structural refinement.
demonstrates consistent improvement over the Boltz-1 baseline, outperforming in terms of both structural accuracy Complex-LDDD) and physical validity across all step budgets.
We acknowledge that our current implementation does not surpass the performance Boltz-2. We attribute this to the underlying base model: our method is fine-tuned using the Boltz-1 architecture, weights, and dataset. contrast, Boltz-2 benefits from significant architectural advancements (e.g., the affinity prediction module) and much larger training dataset particularly focused protein complex interactions. This advantage evident the interface metrics where Boltz-2 dominates.
However, our Gauss-Seide projection module is theoretically agnostic to the base model. To val- idate this potential without the computational cost full fine-tuning, we applied our module as post-processing step on top of Boltz-2. Results in Fig. 7 show that this hybrid approach effec- tively ensures 100% physical validity: it resolves invalid outputs observed in Boltz-2 at low step counts (<10) while maintaining the high structural accuracy of the original model. We thus treat our method as a general purpose physical validity enforcement module that can be flexibly inte- grated into various liffusion-basee structure prediction frameworks.
eererrl-purpose
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Boltz-1 PDB 7X3G
Boltz-1- Beetz---Steering
Boltz-2
Protenix
Protenix-Mini
Ours
PDB 3862
BDBBD
Figure 9: Additional qualitative comparisons. The red color highlights physically invalid struc- tural predictions, such as side chain clashes (PDB 7XSG and PDB 8FJD) and ligand-related clashes (PDB 5SB2). Our approach consistently enforces physical validity.
E COMPUTATIONAL COST OF THE SEIDEL PROJECTION MODULE
We systematically evaluated the computational overhead of incorporating the Gauss-Seidel projec- tion module during both training and inference. To provide canonical measure of input complexity, we benchmarked the module on the CASP15 dataset, binning samples by token count (256, 384, 512, and 640). We reported the forward and backward wall-clock runtimes and additional memory usage on an NVIDIA A100 GPU.
As illustrated Fig. 6 both forward and backward passes are completed within matter seconds, and the additional memory overhead remains below 3.5 GB, even for the largest token bins. This high efficiency is achieved through our specialized implementation, which leverages custom CUDA kernels optimized for both the forward and backward processes.
Table 3: L-RMSD < 2 on the PoseBusters dataset computed by PXMeter (a et al. 2025). Dark green cells denote the best results among the few-step methods. Light green cells denote the best results among the 200-step methods.
Protenix-Mini
Protenix-Mini
Dataset
AlphaFildd
Bolty-1
Protenix
1oltzci-Steering
Boltz-2
10-step
5-step
Ours
PoseBusters
0.86:2:.04
0.71::0.05
0.80:20005
0.71+:0.05
0.91 0.003
0.76::0.05
0.72-L0.05
0.72-:0.05
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F CLARIHICATION ON DOCKING-RELATER METRIC COMPUTATION
In our primary evaluation, calculated docking-related metrics (DockQ > 0.23, and L-RMSD < 2Å) using OpenStructure (Biasini al. 2013), strictly adhering to the protocol defined in the original Boltz-1 paper Wohlwend et al. 2024). We attribute the discrepancies between our reported values and those in the Protenix technical report (Team et al.) 2025) to their use of a different evalua- tion tool, PXMeter (Ma et al. 2025). To investigate this, we performed an evaluation of the L-RMSD < 2Å metric using PXMeter on the PoseBusters dataset for all methods (detailed in Table 3). This re-evaluation confirmed
key points: 1) The re-calculated performance of the Protenix baseline aligns closely with the figures in their original report, validating the source of the numerical dif- ference. While the absolute values depending on the tool used, the relative ranking among the baselines remains unchanged. These results indicate that our comparative analysis yis is informative.
G ANALYSIS OF STRUCTURAL PERTURBATIONS INTRODUCED BY GAUSS-SEIDE PROJECTION
We quantified the structural changes introduced by Gauss-Seidel projection by computing the RMSD between pre- and post-projection structures the PoseBusters dataset. The observed aver- age RMSD 0.1865 + 0.0219 A, confirming that the projection step introduces atomic position changes sufficient to enforce physical validity. Fig. visualizes four representative cases where no- ticeable improvements occurred. For PDB 7P4C, the projection restored a distorted a-helix that was originally clashing with a neighboring strand. In PDB 7PL1, a self-entanglement within a coil re- gion successfully disentangled. Similarly, for PDB 7ELT and PDB 8AEM, the module resolved ligand-siden higand-iidechain and intra-ligand steric clashes. These examples demonstrate that the Gauss-Seidel projection effectively enforees physical validity while preserving the structural prediction of the generative model.
H TRAINING AND INFERENCE ALGORITHMS
We provide pseudo-code for the training and inference algorithms of our model, shown Algo- rithms and The values of hyperparameters shown in both algorithms follow (Gong et al. 2025)
Algorithm Training with Gauss-Seide Projection
Require:
Training dataset consisting of feature-coordinate pairs, D ((f", x(arget)), optimizer O, number of epochs E, number of cycles Neyole. noise co, noise scale X
1: for epoch = 1, ... do
2: for each batch (ff² ,, {xtargery 1) E D do {ssupuss } ATOM ATTENTION ATTENTIONEECCODR(((( ENCODER((f*))
4:
Initialize {st]; {zih} from {s!mpuss
5: for [1, [1,-.-> Neycle] do
{ } f- MSA MfDULE(( fzzz {s inpots
6:
7:
{sa) {zi + PAIRFORMER MODULE((s:), {2zz) 1)
8:
end for
9:
Sample noise: E ((OO,Nxxa)
10:
Construct noisy input: naissy cD x larget + X-
11:
[x] = DIFFUSION MODULE({S)))) (xxoiiy }, Co. (f*), (nputss {si). (21)))
12:
[xmeed] = -EIUEL PROJECTION MODULE(($})
Compute loss: {C} DIFFUSION Loss(( {xtarget
14:
O.zero O.zero_grad()
Backpropagate: [.backward()
16:
Update parameters: O.step()
17:
end for
18:
end for
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7IPU
7XRX
XSS3
7Y15
7Y4H
8B5W
8BBR
BBRY
BBVI
8CLZ
8CR7
8DAJ
8GON
8G53
8GPP
BLB
BQE
8K5K
8OWF
8SUT
Figure 10: Visual comparison with the ground truth. We visualize predicted protein structures together with their corresponding ground truth (pink colored) across a diverse set PDB entries. Each protein's 's name is indicated in the top-left corner. Our method shows consistently close agree- ment with the ground truth, capturing the overall structural features with high fidelity.
I PROOF FOR THE CONVERGENCE OF LINEARIZATION
Theorem I.1 Consider the nonlinear coupled system
-VC() X(x)= =0 a>0,
+xxx)
where C: NN3 R. is twice continuously differentiable. Let the linearized iteration be
I
VC(x(())
Ax]
VC(((())
AA
((x(m)) a (m))
with updates (n+1)) = ((1)) + Ax and X (m+1) = (m) + AX, and initial values x( = &, X (0) = 0.
Then the linearized iterates (x(1)), (n)) converge solution (x"x"") of the nonlinear system.
Proof. Define
gxx, A) (x
VC(x) X,
h(x, A) C(x) +ad,
so that the update of Eq. is equivalent to solving the nonlinear system {g 0, h 0}. Lineariz- ing this system at given point (x(1), ((t) yields the Newton subproblem
K
-C(((()) T
Axi
(g(((), A(0
[C(((()
AA
h(x()) (
a
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Algorithm 4 Inference with Gauss-Seidel Projection
Require: Feature { number of cycles Neycle. noise schedule (c,,Cr, C1, o2), Yos Twin noise scale X,
step scale y)
1: {s + ATOM_ATTENTO (ENTODER_E }{
2: Initialize {8i], (ziz} from {s, [s..pus]
3: for I [1,-.., Neyctel do
4: {21 / < MSA ff
1. {zy) ). a (mputss })
5: {8i) {2iz) + PAIRFORMER MODULE((s) (; (2
6: end for
7: x ~ Co .N(O, TNxa)
8: for Cy [cI, 02] do
2-step diffusion
9:
{x] + CENTRE CENTRE_RANDOM RAADOOAAANNDEEENEEEE AUMMMMTATTTNTNININNI...
= % if C > Yinin else
11:
= (
= = 1v - (N(0,
12:
13:
X noisy x
14:
{x denoiseed } = MOTUUIION_MOULL(((S J. (1*), } (6, (mputss (8,), {z1)))
(x xdencised). 3/8
15:
16:
dl -i
17:
xnoisy + 1..))dl. - f-2 &
18:
19:
end for
20:
(Xproj) GAUSS-SEIDER PRO/EEETONNOOICNNN_LML
21:
return [Xproj)
with K -IEE,,CC,, We thus solve, at each iteration, linear system obtained by Newton linearization of the coupled equations. By Newton's local convergence theorem (Sauer) 2011), the iterates converge quadratically given the Jacobian at the solution (x". 'x) is non-singular and the initialization lies in its basin of attraction. Taking the Schur complement (Zhang 20055) with respect to a > 0 shows that the non-singularity of the Jacobian is equivalent to the invertibility of
S s (I-I (()))(((((x)) - -(-VC(x)) -VV(x=
For sufficiently small yet positive definite a, the term a comes strictly positive definite under standard regularity (full row rank of VC x*), henee invert- ible. Consequently, the Jacobian is non-singular at the solution, and Newton's 's method is guaranteed to converge locally. In practice, the predictor's output x (0)= is close to x", so the initialization within the convergence neighborhood.
dominates and S be-
We then introduce two controlled approximations to avoid forming the Hessian terms in K. First, replace by the identity (i.e., quasi-Newton approximation), which omits the Hessian terms and introduces a local error of order O[] Ax| while retaining a positive-definite S. This does not affect the global eтror or the solution of the fixed-point iteration (Macklin Müller (016). Second, and that gx)(f), ( remains small when constraint gradients vary (Goldenthal et al. 2007: Macklin & Müller 2016 Under these approximations, the linear subproblem reduces to Eq.
we note that g() X (0))
slowly: accordingly, we g(x(f), A
J PROOF FOR THE CONVERGENCE AND CONSTRAINT SATISFACTION OF GAUSS-SEIDEL PROJECTION
Theorem J.1 Consider the linear system
(VC((()))VC(("1") + C(x(m)) a1 0.
(10)
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with each Seuse- update
(1) (x - a,2,
(m)
A
j= L..,, m,
(11)
VC ta,
at the n-th Causs-Seides sweep. After sufficient number of Gauss-Seidel sweeps and sufficiently small a. the iterates converge quasi-second--n manner to the unique solution AX" More-
over, each individual update monotonically reduces the residual error and requires only O(1) work, so a full sweep over all constraints costs (()).
Proof. Define
= VC VV(((")VCCX((") +a, b= -=x((x())) (m)
The solution of Eq. 10 corresponds exactly to the minimizer of the quadratic optimization problem:
min S(AA) 1 A
(12)
2X
With a > 0, is symmetric positive definite (SPD). Hence, Eq. 12 convex, and Eq. 10 has unique solution.
A cyclic Seuus- sweep on Eq. [0 is exactly cyclic coordinate descent on Eq, 12 each update minimizes f w.r.t. single coordinate AAj while holding the others fixed, yielding the closed form in Eq. [1 Because f is strongly convex and quadratic, each coordinate step strictly decreases f unless already optimal, and the sequence (S(A (() monotonically decreasing. For SPD systems, both classical matrix-splitting analysis and coordinate-descent theory guarantee linear convergence of cyclic Gauss-Seided Seissel to the unique minimizer (Vargal 1962 Saad 2003; Wright 2015). In particular, there exists E (0, 1) such that A (k) A A S p AX" Il... Each coordinate update accesses only the atoms affected by the current constraint, so is O(1). A full sweep is thus (mm).
The proof of Theorem I.I shows that Eq. I0 quasi-Newton approximation of first-order Hence, the convergence speed of the Gauss-Seidel projection is of quasi-second order. With striet physical tolerance and sufficiently small penalty parameter a, the final iterate satisfies the optimality conditions of the penalized projection, meaning that in practice it realizes a hard-constrained projection (all constraints are strictly satisfied) (Boyd & Vandenberghe 2004).
optimality condition Eq. S
K PROOF FOR THE CONVERGENCE OF CONIUGATE GRADIENT
Theorem K.1 Consider the adjoint system
T
(x +1)
(13)
@xpros
where
H(x) = 20j [ [
(14)
3=1
Near feasibility, the conjugate gradient method converges when applied to this system.
Proof. At the projected point Xproj- we have C(Xpruj) 0. Hence the second-order small term C, (Xpro) V²C, (Xproj) vanish, and
H (Xpproj)
= Σ 0 (Xpro)) VC,
j=1
This matrix is SPD, and so is H(Xproj) + The conjugate gradient method is guaranteed to con- verge for any linear system with SPD coefficient matrix (Sauer 2011). and its convergence rate is governed by the condition number of the matrix. Since the coefficient matrix contains the identity I, the condition number is bounded below by the size of I, i.e., m, which ensures that the convergence of conjugate gradient remains stable in the neighborhood of feasibility.
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As
xn
x
X,
X;
(a) Cetadi
(b) Cahiral
(c)
Cateren
(d) Cylarer
(e) Crrum
Covuriap
(g)
Ceovaleus
Figure 11: Visualization of the physical constraints used in our framework. Red highlights indicate the structural geometric elements subject to constraints, such as spatial arrangements and interatomic distances, while blue highlights denote double bonds or covalent linkages between atoms.
L DETAILED FORMULATION OF PHYSICAL VALIDITY CONSTRAINTS
Below present the formalization of each of the physical constraints following Boltz-1- Steering [Wohlwend et al. 2024), which are visualized in Fig.
To prevent steric clashes, we impose a constraint on the distance between atoms in distinct, non-bonded chains, requiring it to be greater than 0.725 times the sum of their Van der Waals radii:
(a) Steric Clash.
Colnsh (x)
Σ
max (0.725(r: + + ||x; x²11.0).
(15)
( chains
where r, denotes the Van der Waals radius of atom
(b) Tetrahedral Atom Chirality. For chiral center Z with four substituents ordered by Cahn- Ingold-Prelog (CIP) priority as A, B, C, D, the center is designated R if the bonds (Z A, Z B, Z C) form right- handed system, and S if they form left-handed system. To ensure that the predicted molecular conformers maintain the correct chirality, we define the constraint based on the improper torsion angles (X1; X2, X3, Z):
Cehiral (x)
Σ
max π
DinedralAngllexxx,x, Xk, xx),
(i..,,,))E chiral sets
(16)
Σ
max + DiheeralAAngle(x, X xi).
(5.jA,,))E chiral sets
(c) Bond Stereochemistry. For double bond 2; Z2, where Z1 has substituents A1, Bi and Zo has substituents A2, B2 arranged in decreasing Cahn-IngolddPrellgg (CIP) priority, the double bond said have E stereochemistry if the higher-priority atoms A₁ and Ag lie on opposite sides the bond, and Z stereochemistry otherwise. To ensure that the predicted molecular conformers maintain the correct stereochemistry, we define the constraint based on the torsion angles (A1, Zı Zz, A2) and (B1, 21, Zz, B2):
Csterco (x) =
Σ
5T max (6
DiheerralAngle(x, Xk, xi),
( stereo sets
(17)
DiiherralAngle(xx Xk, xi) π
+
max
() steren sets
(d) Planar Double Bonds. For planar double bonds C1 = C2 between carbon atoms, where C has substituents A,, B, and C2 has substituents A2, Bz, define a flat-bottom constraint based on the improper torsion angles (A1, B2, C1) and (A2, B2, Ci, C2) to enforce planarity of the double bond:
Oplanar (x)
max (DihedralAngll(x,, X,, X4. xi)
1 0).
(18)
(H.J,A.1)E
Strgonal Stignal
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(e) Internal Geometry. To ensure that the model generates ligand conformers with physically realistic distance geometry, we define flat-bottomed constraint based on the bounds matrices gen- erated by the RDKit package:
Cgeom (x) = Σ
max(
Xj 1.20,j, 0) + max(0.8Lx) - ||x: -x,ll.00)
(ood)
Σ
max [x; X,ر1 1.20,;, 0) max(0.8L.;; |x; -x,11,0)
(19)
max([|x; - xyy1 -x11
1.201;, + max(0.8L.) - ||x; - x, 0).
(3.3) USgge
where and U E NNN denote the lower and upper bounds matrices. For a pair of atoms (i.j). Lij gives the lower distance bound and ULj gives the upper distance bound (Buttenschoen et al. 2024
(f) Overlapping Chains. To prevent overlapping chains, we define constraint based the dis- tance between the centroids of symmetric chains with more than one atom:
Coverlap (x)
max (1.0- [ xgl1.0)
(20)
(A,B)E S.ymaaSimn ctates
where A is the centroid of chain A.
(g) Covalently Bonded Chains. To ensure the model respects covalently bonded chains, we define a constraint enforce that covalently bonded atoms between separate chains are within 2 2A:
Ceovalent (x)
max (|xx; -x,1-2,00) 2.0)
(21)
(
M DETAILS ON BASELINES AND EVALUATION PROTOCOL
We compare method against strong baselines: Boltz-1 (the original model) (Wohlwend jet al. 2024), Boltz-1-Steering (Boltz-1 with FK-steering at inference), Boltz-2 (the updated Boltz model) (Pasarro et al. 2025). Protenix (Team et al. (2055) and Protenix- (Gong al. 2025) All baselines use 200-step diffusion sampling, whereas Protenix-Mini uses a 5-step sampling. We evalu- our method and all baselines six benchmarks: CASP15, Test, PoseBusters, AF3-AB, dsDNA, and RNN--Potein. CASP15 and Test (from Boltz-1 (Wohlwend et al.) 2024)) cover diverse assem- blies, including protein complexes, nucleic acids, and small molecules. PoseBusters (Buttenschoenl et al. 2024) focuses on protein-ligand modeling. AF3-AB (released with AlphaFold3 (Abramson aantibdy complexes. dsDNA (Ma et al. 2025) contains entries with two DNA chains and one protein chain. RNA-Protein (Ma et al. 2025 contains entries consisting of one RNA chain and one protein chain.
al. 2024)) contains
We compare our method with baselines using standard evaluation metrics, including Complex LDDT, Prot-Prot iLDDT, Lig-Prot iLDDT, DNA-Prot iL.DTT, RNA-Prot iLDDT, DockQ > 0.23, L-RMSD 2 A, TM-score, and Physical Validity. Complex LDDT measures the overall geomet- ric accuracy throughout the complex, while Prot-Prot iLDDT, Lig-Prot iLDDT, DNA-Prot iLDDT, and RNA-Pro Prot iLDDT focus specifically on the quality of protein-proteins
pattii-llggand protein- DNA, and protein-RNA interfaces, respectively. The average DockQ success rate (DockQ > 0.23), defined as the proportion predictions with DockQ > 0.23, measures the fraction of cases where good protein-protein interactions are correctly predicted. L-RMSD is defined as the proportion of ligands with a pocket-aligned RMSD below 2 Å, widely adopted measure of molecular docking accuracy. TM-score is a length-normalized measure of structural similarity, widely used to as- sess protein structure prediction accuracy. Finally, Physical Validity checks whether the predicted structures obey fundamental physical constraints, including steric clash, tetrahedral atomic chirality, bond stereochemistry, planar double bonds, internal geometry, overlapping chains, and covalently bound chains. All evaluation metrics follow those used in Boltz-1 (Wohlwend et al., 2024) and Protenix Team al. 2025
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N
THE USE OF LARGE LANGUAGE MODELS (LLMS)
Consistent with the conference policy on LLM usage, we used LLMss solely for language polishing and improving the clarity of presentation. In particular:
Scope of use. LLMs were used for language polishing (rephrasing sentences, tightening tran- sitions, standardizing terminology). They were not used for research ideation, experimental de- sign, data analysis, derivations, algorithmic contributions, code figure generation, literature search/screening, or drafting technical content beyond author-provided prose.
Authorship and responsibility. All scientific claims, methods, proof', experiments, and conclu- sions are authored by the listed authors. We carefully reviewed and verified any text edited with LLM assistance. The authors accept full responsibility for the contents of the paper, including any edited passages, and for ensuring the absence of plagiarism fabricated material.
Attribution and citations. All references and technical statements were selected, verified, and cited by the authors; LL.Ms were not used to generate or suggest citations.
Data and privacy. No proprietary, confidential, personally identifiable information pro- vided to LLM tools.
24


Reviews:
[{"review_id": "cnJCzPjRoT", "reviewer": "Reviewer_YVMA", "summary": "This paper presents a Gauss–Seidel–based projection module that enables stable and rapid convergence to physically plausible structures using only a few denoising steps. By explicitly enforcing geometric constraints during sampling, the method significantly accelerates inference while maintaining a high degree of physical validity.", "questions": "See weakness", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The use of a Gauss–Seidel projection mechanism to enforce physical validity is novel and appears highly effective. It provides a computationally efficient way to ensure that generated structures adhere to key geometric constraints.\n\n2. Both quantitative and qualitative results demonstrate that the proposed method is effective and efficient across various evaluation metrics.", "weaknesses": "1. The paper does not release its code, project page, or any runnable repository. This raises concerns regarding reproducibility and the ability of other researchers to validate or extend the results.\n\n2. The experiments are conducted by finetuning the method on an existing co-folding model (Boltz-1). However, the reported two-step sampling results do not consistently surpass the original Boltz-1 baseline. Although this may be attributable to the reduced number of denoising steps, it remains unclear whether the proposed method can outperform Boltz-1 when using a comparable number of steps.\n\n3. The comparison with Proteinix-mini appears somewhat unfair, as Proteinix-mini uses a significantly reduced architecture and removes the MSA module, placing it at an inherent disadvantage.\n\n4. Since the second-stage projection refines the output from the first denoising step, the method improves physical validity but may have limited capacity to further enhance structural accuracy. It is unclear whether the approach can simultaneously improve both accuracy and validity, or whether it inherently trades one for the other.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "w0eJUY49OA", "reviewer": "Reviewer_66sZ", "summary": "This paper introduces a physically constrained diffusion framework for biomolecular structure generation that enforces local geometric and energetic constraints while maintaining structural accuracy. The method integrates a Gauss–Seidel based differentiable projection that iteratively enforces constraints to ensure the physical feasibility of generated structures. The framework supports backpropagation through conjugate gradients, enabling stable gradient-based training and inference. Empirical results demonstrate improved geometric validity, structural stability, and fast inference across complex biomolecular benchmarks.", "questions": "1. Since the Gauss–Seidel scheme guarantees uniqueness only for each linearized subproblem, can different orderings or initialization seeds lead to non-unique final projections? Have the authors observed multiple feasible solutions in practice?\n2. The drop in docking-related metrics on PoseBusters is notable. Can the authors provide analysis or ablation evidence on whether this degradation stems from the specific constraint choices or the limited number of denoising steps?\n3. Why is metrics such as iLDDT excluded from Table 2? Including complementary metrics such as TM-Score or PoseBusters-valid success rate would make the evaluation more comprehensive.\n4. Could the authors clarify how $\\alpha$ is chosen in practice, and whether convergence depends sensitively on this value?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The motivation is clear and well grounded in the need for physically valid biomolecular generation.\n- The method enforces atomically realistic outputs compared to unconstrained baselines, a practically important contribution.\n- The algorithmic formulation is sound. The iterative Gauss–Seidel projection and penalty method are mathematically principled and differentiable.", "weaknesses": "- The PoseBusters results show a notable drop in docking-related metrics, but the paper does not analyze the cause, possibly due to a trade-off between hard constraint enforcement and ligand flexibility.\n- Some evaluation metrics are missing. Including measures such as the PoseBusters-valid success rate, TM-Score, and iLDDT (in Table 2) would make the empirical validation more complete and convincing.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "vuVQLQQnAs", "reviewer": "Reviewer_3NuJ", "summary": "This paper tackles the critical issue of physical invalidity (steric clashes, distorted geometry) in structures generated by deep learning models for biomolecular interactions, particularly all-atom diffusion models. The authors propose a method to enforce physical validity as a hard constraint during both training and inference. The core idea is a differentiable projection module that maps the provisional atomic coordinates generated by a diffusion model onto the nearest physically valid configuration . This projection is efficiently implemented using a Gauss-Seidel iterative scheme, exploiting the locality of physical constraints . Crucially, the module is made differentiable via implicit differentiation, allowing it to be integrated seamlessly into existing diffusion frameworks (like Boltz-1) for end-to-end finetuning . A key result is that incorporating this module enables the generation of physically valid and structurally accurate complexes using only 2 denoising steps, achieving accuracy comparable to 200-step SOTA baselines while offering ~10x speedup and guaranteeing validity . The method is evaluated on six diverse benchmarks against strong baselines.", "questions": "- Can the authors provide metrics (e.g. RMSD) quantifying the magnitude of coordinate changes introduced by the projection step? Are there cases where projection significantly alters key interface interactions or secondary structure elements?", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- Generating physically plausible structures is a prerequisite for the reliability and utility of biomolecular models. This paper directly confronts the common failing of deep generative models in this regard , offering a principled solution.\n- The use of the Gauss-Seidel method for the projection step is well-suited for the problem, leveraging the local nature of physical constraints (bond lengths, angles, clashes) for fast and stable convergence compared to methods like gradient descent.\n- Making the iterative Gauss-Seidel solver differentiable via implicit differentiation  is a key technical contribution, enabling end-to-end training and allowing the diffusion model to adapt to the projection step. This integration is crucial for maintaining high accuracy, as shown in the ablation study.", "weaknesses": "- The evaluation of the Protenix baseline appears to be an underestimation. According to the Protenix technical report, as well as anecdotal user feedback, its performance at 200 steps (e.g., in terms of DockQ and validity metrics) is reportedly not as low as presented in this paper. I recommand the authors to either provide the detailed configuration (config) files used for their Protenix experiments or, preferably, release the raw prediction files generated by their baseline models to ensure a fair and reproducible comparison.\n- Regarding the Protenix mini tech report, the original authors claim that 'increasing the ODE sampling steps to 10 effectively mitigates this issue (clash)'. Given this, the comparison in Table 1 might be suboptimal. I strongly suggest that the authors re-evaluate the baseline using 10 ODE steps in Table 1, as this seems to be the recommended setting for mitigating structural clashes.\n- The paper states that GS projection achieves good physical constraint effects within just 2 steps. This raises a question: why did the authors not experiment with applying the projection for more steps? Furthermore, the results in Table 1 indicate that sampling for 2 steps with constraints still underperforms the original 'boltz2' baseline. The comparison is incomplete. The authors should also include experiments comparing their method against baselines like 'boltz2' when restricted to a similar small step budget (e.g., 10 steps) for structure prediction.\n- Although the 2-step process is much faster overall, the paper could provide more details on the computational cost (time and memory, forward and backward) of the Gauss-Seidel projection module itself, especially as the system (complex) size grows larger. The backward pass involves solving a linear system using CG, which could become expensive.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "vJ58aXGQo3", "reviewer": "Reviewer_iHkC", "summary": "This paper addresses the physical validity challenge in generative modeling for protein structures. It proposes a first-order constraint and enforces it during both fine-tuning and inference processes. The constraint is formulated as an optimization problem, which is approximately solved using Gauss-Seidel updates. Additionally, the paper introduces implicit differentiation to enable backpropagation during fine-tuning. Compared to baseline methods, this approach achieves faster inference, requiring only 2-step denoising, while maintaining competitive accuracy.", "questions": "1. In Equation (2), Is there a redundant factor of 0.5 in the definition of E(x)? It seems to be inconsistent with Equation (3).\n2. The iterative linearization of eq.4 (Theorem D.1) relies on multiple approximations (replacing K by I; assume g = 0). Could you please discuss how these approximations affect the convergence speed and solution quality of the optimization problem? Maybe by numerical simulations or tests on real protein datasets) could help validate the affect of these approximations.\n3. Could there be additional textual description to clarify the \"invalid\" structures in Figure 4? It is not always obvious which structural issues are presented: Are all cases attributed to atomic clashes, or are there other problems (e.g., bond length violations, steric hindrance)?\n4. How does the method perform without relying on the Protenix-mini sampling strategy and compare it to Boltz-1?\n5. While the paper emphasizes reducing the number of sampling steps from 5 to 2, it is unclear why this results in a wall-clock time reduction significantly greater than 2.5× in Figure 5 (right) when compared to Protenix-mini.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The manuscript is well-written, and easy to follow.\n2. The performance in terms of inference acceleration is impressive.\n3. The proposed method is reasonable and offers valuable insights to the field of AI4S. In many scientific applications, strict physical laws must be upheld. When training data is limited, models often struggle to learn these constraints in a data-driven way; thus, explicitly enforcing the constraint is a reasonable and effective solution.", "weaknesses": "1. While the main focus of the paper is on inference performance under a small number of sampling steps, I believe it would be beneficial to also include results under larger-step settings. This would provide a more comprehensive understanding of the method's performance when the computational budget is less constrained, and offer a clearer comparison with existing approaches in such scenarios.\n2. It would be beneficial to discuss the relationship between proposed method with reinforcement learning (RL)-based fine-tuning approaches for diffusion models (e.g., DPOK, DRaFT). Compared to inference time guidance methods, they also involve fine-tuning to optimizing the reward. I’m curious about how the fine-tuning cost of RL compares to that of your iterative optimization algorithm.\n3. How is the scalability of your method? Since your method involves enforcing constraints on all substructures, as the complexity or size of the system grows, does the method remain computationally feasible during training and inference?\n4. Alphafold3 is an important baseline in this domain and is suggested to be included in experiment section for a more complete comparison. Additionally, the evaluation could benefit from incorporating more comprehensive metrics such as TM-score and pb-valid.\n5. To facilitate reproducibility and enable further research based on this work, I strongly encourage the authors to release the code (such as in an anonymous form during the review period).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)