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Applications · Physics

Zhiyuan Wang, Jinwoo Go, Byung-Jun Yoon, Nathan Urban, Xiaoning Qian

In recent developments in scientific machine learning (SciML), neural surrogate solvers for partial differential equations (PDEs) have become powerful tools for accelerating scientific computation for various science and engineering applications. However, training neural PDE solvers often demands a large amount of high-fidelity PDE simulation data, which are expensive to generate. Active learning (AL) offers a promising solution by adaptively selecting training data from the PDE settings--including parameters, initial and boundary conditions--that are expected to be most informative to help reduce this data burden. In this work, we introduce PaPQS, a Plug-and-Play Query Synthesis AL framework that synthesizes informative PDE settings directly in the continuous design space. PaPQS optimizes the Expected Information Gain (EIG) while encouraging batch diversity, enabling model-aware exploration of the design space via backpropagation through the neural PDE solution trajectories. The framework is applicable to general PDE systems and surrogate architectures, and can be seamlessly integrated with existing AL strategies. Extensive experiments across different PDE systems demonstrate that our AL framework, PaPQS, consistently improves sample efficiency over existing AL baselines.

Computer Vision · 3D Rendering & Reconstruction

Junying Wang, Yuanlu Xu, Edith Tretschk, Ziyan Wang, Anastasia Ianina, Aljaz Bozic, Ulrich Neumann, Tony Tung

The creation of photorealistic dynamic hair remains a major challenge in digital human modeling because of the complex motions, occlusions, and light scattering. Existing methods often resort to static capture and physics-based models that do not scale as they require manual parameter fine-tuning to handle the diversity of hairstyles and motions, and heavy computation to obtain high-quality appearance. In this paper, we present Dynamic Gaussian Hair (DGH), a novel framework that efficiently learns hair dynamics and appearance. We propose: (1) a coarse-to-fine model that learns temporally coherent hair motion dynamics across diverse hairstyles; (2) a strand-guided optimization module that learns a dynamic 3D Gaussian representation for hair appearance with support for differentiable rendering, enabling gradient-based learning of view-consistent appearance under motion. Unlike prior simulation-based pipelines, our approach is fully data-driven, scales with training data, and generalizes across various hairstyles and head motion sequences. Additionally, DGH can be seamlessly integrated into a 3D Gaussian avatar framework, enabling realistic, animatable hair for high-fidelity avatar representation. DGH achieves promising geometry and appearance results, providing a scalable, data-driven alternative to physics-based simulation and rendering.

Applications · Physics

Ziyu Xiang, Kenna Ashen, Xiaofeng Qian, Xiaoning Qian

Symbolic regression (SR) seeks interpretable analytical expressions that uncover the governing relationships within data, providing mechanistic insight beyond 'black-box' models. However, existing SR methods often suffer from two key limitations: (1) *redundant representations* that fail to capture mathematical equivalences and higher-order operand relations, breaking permutation invariance and hindering efficient learning; and (2) *sparse rewards* caused by incomplete incorporation of constraints that can only be evaluated on full expressions, such as constant fitting or physical-law verification. To address these challenges, we propose a unified framework, **Graph-based Symbolic Regression (GSR)**, which compresses the search space through a permutation-invariant representation, Expression Graphs (EGs), that intrinsically encode expression equivalences via a term-rewriting system (TRS) and a directed acyclic graph (DAG) structure; and mitigates reward sparsity via employing hybrid neural-guided Monte-Carlo tree search (hnMCTS) on EGs, where the constraint-informed neural guidance enables direct incorporation of expression-level constraint priors, and an adaptive $\epsilon$-UCB policy balances exploration and exploitation. Theoretical analyses establish the uniqueness of our proposed EG representation and the convergence of the hnMCTS algorithm. Experiments on synthetic and real-world scientific datasets demonstrate the efficiency and accuracy of GSR in discovering underlying expressions and adhering to physical laws, offering practical solutions for scientific discovery.

Theory · Learning Theory

Weikuo Wang, Yue Liao, Huan Luo

A major problem of kernel-based methods (e.g., least squares support vector machines, LS-SVMs) for solving linear/nonlinear ordinary differential equations (ODEs) is the prohibitive $O(an^3)$ ($a=1$ for linear ODEs and 27 for nonlinear ODEs) part of their computational complexity with increasing temporal discretization points $n$. We propose a novel Nyström-accelerated LS-SVMs framework that breaks this bottleneck by reformulating ODEs as primal-space constraints. Specifically, we derive for the first time an explicit Nyström-based mapping and its derivatives from one-dimensional temporal discretization points to a higher $m$-dimensional feature space ($1< m\le n$), enabling the learning process to solve linear/nonlinear equation systems with $m$-dependent complexity. Numerical experiments on sixteen benchmark ODEs demonstrate: 1) $10-6000$ times faster computation than classical LS-SVMs and physics-informed neural networks (PINNs), 2) comparable accuracy to LS-SVMs ($<0.13\%$ relative MAE, RMSE, and $\left \| y-\hat{y} \right \| _{\infty } $difference) while maximum surpassing PINNs by 72\% in RMSE, and 3) scalability to $n=10^4$ time steps with $m=50$ features. This work establishes a new paradigm for efficient kernel-based ODEs learning without significantly sacrificing the accuracy of the solution.

Computer Vision · Vision Models & Multimodal

Kun Xiang, Heng Li, Terry Jingchen Zhang, Yinya Huang, Zirong Liu, Peixin Qu, Jixi He, Jiaqi Chen, Yu-Jie Yuan, Jianhua Han 等

We present SeePhys, a large-scale multimodal benchmark for LLM reasoning grounded in physics questions ranging from middle school to PhD qualifying exams. The benchmark covers 7 fundamental domains spanning the physics discipline, incorporating 21 categories of highly heterogeneous diagrams. In contrast to prior works where visual elements mainly serve auxiliary purposes, our benchmark features a substantial proportion of vision-essential problems (75%) that mandate visual information extraction for correct solutions. Through extensive evaluation, we observe that even the most advanced visual reasoning models (e.g., Gemini-2.5-pro and o4-mini) achieve sub-60% accuracy on our benchmark. These results reveal fundamental challenges in current large language models' visual understanding capabilities, particularly in: (i) establishing rigorous coupling between diagram interpretation and physics reasoning, and (ii) overcoming their persistent reliance on textual cues as cognitive shortcuts.Project Page: github.com/SeePhys/seephys-projectHugging Face: huggingface.co/datasets/SeePhys/SeePhys

Applications · Physics

Yunfeng Liao, Yangxin Wu, Xiucheng Li

Recent advances in graph neural network (GNN)-based neural operators have demonstrated significant progress in solving partial differential equations (PDEs) by effectively representing computational meshes. However, most existing approaches overlook the intrinsic physical and topological meaning of higher-order elements in the mesh, which are closely tied to differential forms. In this paper, we propose a higher-order GNN framework that incorporates higher-order interactions based on discrete and finite element exterior calculus. The time-independent boundary value problems (BVPs) in electromagnetism are instantiated to illustrate the proposed framework. It can be easily generalized to other PDEs that admit differential form formulations. Moreover, the novel physics-informed loss terms, integrated form estimators, and theoretical support are derived correspondingly. Experiments show that our proposed method outperforms the existing neural operators by large margins on BVPs in electromagnetism. Our code is available at https://github.com/Supradax/Higher-Order-Differential-Topology-aware-GNN.

General Machine Learning · Causality

Benjamin Herdeanu, Juan Nathaniel, Carla Roesch, Jatan Buch, Gregor Ramien, Johannes Haux, Pierre Gentine

Causal discovery for dynamical systems poses a major challenge in fields where active interventions are infeasible. Most methods used to investigate these systems and their associated benchmarks are tailored to deterministic, low-dimensional and weakly nonlinear time-series data. To address these limitations, we present *CausalDynamics*, a large-scale benchmark and extensible data generation framework to advance the structural discovery of dynamical causal models. Our benchmark consists of true causal graphs derived from thousands of both linearly and nonlinearly coupled ordinary and stochastic differential equations as well as two idealized climate models. We perform a comprehensive evaluation of state-of-the-art causal discovery algorithms for graph reconstruction on systems with noisy, confounded, and lagged dynamics. *CausalDynamics* consists of a plug-and-play, build-your-own coupling workflow that enables the construction of a hierarchy of physical systems. We anticipate that our framework will facilitate the development of robust causal discovery algorithms that are broadly applicable across domains while addressing their unique challenges. We provide a user-friendly implementation and documentation on https://kausable.github.io/CausalDynamics.

Applications · Physics

Andrew F. Ilersich, Prasanth Nair

The physical sciences are replete with dynamical systems that require the resolution of a wide range of length and time scales. This presents significant computational challenges since direct numerical simulation requires discretization at the finest relevant scales, leading to a high-dimensional state space. In this work, we propose an approach to learn stochastic multiscale models in the form of stochastic differential equations directly from observational data. Drawing inspiration from physics-based multiscale modeling approaches, we resolve the macroscale state on a coarse mesh while introducing a microscale latent state to explicitly model unresolved dynamics. We learn the parameters of the multiscale model using a simulator-free amortized variational inference method with a Product of Experts likelihood that enforces scale separation. We present detailed numerical studies to demonstrate that our learned multiscale models achieve superior predictive accuracy compared to under-resolved direct numerical simulation and closure-type models at equivalent resolution, as well as reduced-order modeling approaches.

Applications · Physics

Haoran Li, CHENHAN XIAO, Muhao Guo, Yang Weng

Learning dynamics is essential for model-based control and Reinforcement Learning in systems operating in changing environments, such as robotics, autonomous vehicles, and power systems. However, limited system measurements, such as those from low-resolution meters, demand sample-efficient learning. Symmetry provides a powerful inductive bias by characterizing equivalent relations in system behavior to improve sample efficiency. While recent methods attempt to discover symmetries from data, they typically assume a single global symmetry group and treat symmetry discovery and dynamic learning as separate tasks, leading to limited expressiveness and error accumulation. In this paper, we propose the Latent Mixture of Symmetries (Latent MoS), an expressive model that captures symmetry-governed latent factors from complex dynamical measurements. Latent MoS focuses on dynamic learning while locally preserving the underlying symmetric transformations. To further capture long-range temporal equivalence, we introduce a hierarchical architecture that stacks Latent MoS blocks across multiple time scales. Numerical experiments across diverse physical systems demonstrate that Latent MoS significantly outperforms state-of-the-art baselines in interpolation and extrapolation tasks while offering interpretable latent representations suitable for future geometric and safety-critical analysis.

Applications · Physics

Seungwoo Yoo, Kyeongmin Yeo, Jisung Hwang, Minhyuk Sung

We introduce Neural Green’s Function, a neural solution operator for linear partial differential equations (PDEs) whose differential operators admit eigendecompositions. Inspired by Green’s functions, the solution operators of linear PDEs that depend exclusively on the domain geometry, we design Neural Green’s Function to imitate their behavior, achieving superior generalization across diverse irregular geometries and source and boundary functions. Specifically, Neural Green’s Function extracts per-point features from a volumetric point cloud representing the problem domain and uses them to predict a decomposition of the solution operator, which is subsequently applied to evaluate solutions via numerical integration. Unlike recent learning-based solution operators, which often struggle to generalize to unseen source or boundary functions, our framework is, by design, agnostic to the specific functions used during training, enabling robust and efficient generalization. In the steady-state thermal analysis of mechanical part geometries from the MCB dataset, Neural Green’s Function outperforms state-of-the-art neural operators, achieving an average error reduction of 13.9% across five shape categories, while being up to 350 times faster than a numerical solver that requires computationally expensive meshing.

Applications

Chaohao Yuan, Maoji Wen, Ercan KURUOGLU, Yang Liu, Jia Li, Tingyang Xu, Deli Zhao, Hong Cheng, Yu Rong

To enhance the generalization ability of graph neural networks (GNNs) in learning and simulation physical dynamics, a series of equivariant GNNs have been developed to incorporate the symmetric inductive bias. However, the existing methods do not take into account the non-stationarity nature of physical dynamics, where the joint distribution changes over time. Moreover, previous approaches for modeling non-stationary time series typically involve normalizing the data, which disrupts the symmetric assumption inherent in physical dynamics. To model the non-stationary physical dynamics while preserving the symmetric inductive bias, we introduce a Non-Stationary Equivariant Graph Neural Network (NS-EGNN) to capture the non-stationarity in physical dynamics while preserving the symmetric property of the model. Specifically, NS-EGNN employs Fourier Transform on segments of physical dynamics to extract time-varying frequency information from the trajectories. It then uses the first and second-order differences to mitigate non-stationarity, followed by pooling for future predictions. Through capturing varying frequency characteristics and alleviate the linear and quadric trend in the raw physical dynamics, NS-EGNN better models the temporal dependencies in the physical dynamics. NS-EGNN has been applied on various types of physical dynamics, including molecular, motion and protein dynamics. In various scenario, NS-EGNN consistently surpasses the performance of existing state-of-the-art algorithms, underscoring its effectiveness. The implementation of NS-EGNN is available at https://github.com/MaojiWEN/NS-EGNN.

Applications · Physics

Hao Wei, Aleksandra Franz, Björn List, Nils Thuerey

When simulating partial differential equations, hybrid solvers combine coarse numerical solvers with learned correctors. They promise accelerated simulations while adhering to physical constraints. However, as shown in our theoretical framework, directly applying learned corrections to solver outputs leads to significant autoregressive errors, which originate from amplified perturbations that accumulate during long-term rollouts, especially in chaotic regimes. To overcome this, we propose the Indirect Neural Corrector ($\mathrm{INC}$), which integrates learned corrections into the governing equations rather than applying direct state updates. Our key insight is that $\mathrm{INC}$ reduces the error amplification on the order of $\Delta t^{-1} + L$, where $\Delta t$ is the timestep and $L$ the Lipschitz constant. At the same time, our framework poses no architectural requirements and integrates seamlessly with arbitrary neural networks and solvers. We test $\mathrm{INC}$ in extensive benchmarks, covering numerous differentiable solvers, neural backbones, and test cases ranging from a 1D chaotic system to 3D turbulence. INC improves the long-term trajectory performance ($R^2$) by up to 158.7\%, stabilizes blowups under aggressive coarsening, and for complex 3D turbulence cases yields speed-ups of several orders of magnitude. INC thus enables stable, efficient PDE emulation with formal error reduction, paving the way for faster scientific and engineering simulations with reliable physics guarantees.

Applications · Physics

Leslie Hwang

Physics-informed neural networks (PINNs) have emerged as a promising framework to develop complex scientific surrogate models, yet their scalability and accuracy often degrade in non-canonical geometries, such as non-rectangular domains or three-dimensional (3D) domains with high aspect ratios. These limitations hinder the broader adoption of vanilla PINNs in real-world, practical systems. In this work, we introduce a multi-domain PINN (MDPINN) framework designed to address the scalability and generalization challenges inherent in 3D non-rectangular domains governed by nonlinear fluid dynamics. The target domain consists of intersecting 3D fluid channels with a high aspect ratio, inducing complex flow features such as deflections, mixing, and recirculations. Our approach is grounded in two key innovations: 1) domain decomposition, which partitions the channel volumes into multiple cubic-like subdomains, each modeled by an individual PINN, 2) enforcement of global dynamics (MDPINN-GD), which ensures that the total mass flow rate entering the domain equals that exiting. These innovations reduce the complexity of the problem imposed on individual PINNs and guide effective network optimization toward physically consistent solutions throughout the domain. We demonstrate that our method achieves: 1) 74.8\% accuracy improvement over a single-network PINN, and 2) 52.9\% accuracy improvement over MDPINN that do not enforce global mass conservation. Furthermore, the MDPINN-GD framework exhibits accurate prediction even in highly complex regions-such as the channel intersecting zone and the outlet zone characterized by intense flow mixing and large velocity gradients-achieving maximum normalized mean absolute errors below 14.9\% for velocity predictions compared to simulation results. This work establishes a path towards scalable, physically grounded surrogate modeling approach that is extensible to multiphysics and high-dimensional scientific problems.

Applications · Chemistry and Drug Discovery

Diya Zhang, Mengwei Sun, Xingdan Wang, Cheng Liang, Qiaozhen Meng, Shiqiang Ma, Fei Guo

Accurately modeling the target-drug complex at atom level presents a significant challenge in the computer-aided drug design. Traditional methods that rely solely on rigid transformations often fail to capture the adaptive interactions between targets and drugs, particularly during substantial conformational changes in targets upon ligand binding, which becomes especially critical when learning target-drug interactions in drug design. Accurately modeling these changes is crucial for understanding target-drug interactions and improving drug efficacy. To address these challenges, we introduce DynaPhArM, an SE(3)-Equivariant Transformer model specifically designed to capture adaptive alterations occurring within target-drug interactions. DynaPhArM utilizes the cooperative scalar-vector representation, drug-specific embeddings, and a diffusion process to effectively model the evolving dynamics of interactions between targets and drugs. Furthermore, we integrate physical information and energetic principles that maintain essential geometric constraints, such as bond lengths, bond angles, van der Waals forces (vdW), within a multi-task learning (MTL) framework to enhance accuracy. Experimental results demonstrate that DynaPhArM achieves state-of-the-art performance with an overall root mean square deviation (RMSD) of 2.01 Å and a sc-RMSD of 0.29 Å while exhibiting higher success rates compared to existing methodologies. Additionally, DynaPhArM shows promise in enhancing drug specificity, thereby simulating how targets adapt to various drugs through precise modeling of atomic-level interactions and conformational flexibility.

Computer Vision · Vision Models & Multimodal

Guanlin (Frank) Wu, Boyan Su, Yang Zhao, Pu Wang, Yichen Lin, Hao Frank Yang

How to integrate and verify spatial intelligence in foundation models remains an open challenge. Current practice often proxies Visual-Spatial Intelligence (VSI) with purely textual prompts and VQA-style scoring, which obscures geometry, invites linguistic shortcuts, and weakens attribution to genuinely spatial skills. We introduce Spatial Intelligence Grid (SIG): a structured, grid-based schema that explicitly encodes object layouts, inter-object relations, and physically grounded priors. As a complementary channel to text, SIG provides a faithful, compositional representation of scene structure for foundation-model reasoning. Building on SIG, we derive SIG-informed evaluation metrics that quantify a model’s intrinsic VSI, which separates spatial capability from language priors. In few-shot in-context learning with state-of-the-art multimodal LLMs (e.g. GPT- and Gemini-family models), SIG yields consistently larger, more stable, and more comprehensive gains across all VSI metrics compared to VQA-only representations, indicating its promise as a data-labeling and training schema for learning VSI. We also release SIGBench, a benchmark of 1.4K driving frames annotated with ground-truth SIG labels and human gaze traces, supporting both grid-based machine VSI tasks and attention-driven, human-like VSI tasks in autonomous-driving scenarios.

Applications · Robotics

Atharva Gundawar, Som Sagar, Ransalu Senanayake

Vision-Language Models (VLMs) are increasingly pivotal for generalist robot manipulation, enabling tasks such as physical reasoning, policy generation, and failure detection. However, their proficiency in these high-level applications often assumes a deep understanding of low-level physical prerequisites, a capability that is largely unverified. To perform actions reliably, robots must comprehend intrinsic object properties (e.g., material, weight), action affordances (e.g., graspable, stackable), and physical constraints (e.g., stability, reachability, or an object's state like being closed). Despite their ubiquitous use in manipulation, we argue that off-the-shelf VLMs may lack this granular, physically-grounded understanding, as these specific prerequisites are often overlooked during training. Addressing this critical gap, we introduce PAC Bench, a comprehensive benchmark designed to systematically evaluate VLMs on their understanding of these core Properties, Affordances, and Constraints (PAC) from a task executability perspective. PAC Bench features a diverse dataset with more than 30,000 annotations, comprising 673 real-world images (115 object classes, 15 property types, 1–3 affordances defined per object class), 100 real-world humanoid view scenarios, and 120 unique simulated constraint scenarios across four tasks. Our evaluations reveal significant gaps in the ability of VLMs to grasp fundamental physical concepts, underscoring their current limitations for reliable robot manipulation and pointing to key areas that require targeted research. PAC Bench also serves as a standardized benchmark for rigorously evaluating the physical reasoning capabilities of VLMs guiding the development of more robust and physically grounded models for robot manipulation.

Applications · Chemistry and Drug Discovery

Yanqiao Zhu, Yidan Shi, Yuanzhou Chen, Fang Sun, Yizhou Sun, Wei Wang

Molecular representation learning has emerged as a promising approach for modeling molecules with deep learning in chemistry and beyond. While 3D geometric models effectively capture molecular structure, they typically process single static conformers, overlooking the inherent flexibility and dynamics of molecules. In reality, many molecular properties depend on distributions of thermodynamically accessible conformations rather than single structures. Recent works show that learning from conformer ensembles can improve molecular representations, but existing approaches either produce unphysical structures through averaging or require restrictive molecular alignment. In this paper, we propose SymmetryPreserving Conformer Ensemble networks (SPiCE), which introduces two key innovations: (1) geometric mixture-of-experts for selective processing of scalar and vector features, and (2) hierarchical ensemble encoding that combines ensemblelevel representation with cross-conformer integration. Crucially, SPiCE ensures physically meaningful representations by maintaining joint equivariance to geometric transformations of individual conformers and conformer permutations. Extensive experiments demonstrate that SPiCE consistently outperforms existing conformer ensemble methods and state-of-the-art structural aggregation models across quantum mechanical and biological property prediction tasks.

Computer Vision · Everything Else

shengchuan gao, Shuo Wang, Yabiao Wang, Ran Yi

This work tackles a challenging problem: stochastic human motion prediction (SHMP), which aims to forecast diverse and physically plausible future pose sequences based on a short history of observed motion. While autoregressive sequence models have excelled in related generation tasks, their reliance on vector‐quantized tokenization limits motion fidelity and training stability. To overcome these drawbacks, we introduce \textbf{WaveAR}, a novel AR based framework which is the first successful application of a continuous autoregressive generation paradigm to HMP to our best knowledge. WaveAR consists of two stages. In the first stage, a lightweight Spatio‐Temporal VAE (ST-VAE) compresses the raw 3D-joint sequence into a downsampled latent token stream, providing a compact yet expressive foundation. In the second stage, we apply masked autoregressive prediction directly in this continuous latent space, conditioning on both unmasked latents and multi‐scale spectral cues extracted via a 2D discrete wavelet transform. A fusion module consisting of alternating cross-attention and self-attention layers adaptively fuses temporal context with low- and high-frequency wavelet subbands, and a small MLP‐based diffusion head predicts per-token noise residuals under a denoising loss. By avoiding vector quantization and integrating localized frequency information, WaveAR preserves fine‐grained motion details while maintaining fast inference speed. Extensive experiments on standard benchmarks demonstrate that our approach delivers more accurate and computationally efficient predictions than prior state‐of-the-art methods.

Applications · Physics

Minchan Jeong, Jongha (Jon) Ryu, Se-Young Yun, Gregory Wornell

The Koopman operator provides a principled framework for analyzing nonlinear dynamical systems through linear operator theory. Recent advances in dynamic mode decomposition (DMD) have shown that trajectory data can be used to identify dominant modes of a system in a data-driven manner. Building on this idea, deep learning methods such as VAMPnet and DPNet have been proposed to learn the leading singular subspaces of the Koopman operator. However, these methods require backpropagation through potentially numerically unstable operations on empirical second moment matrices, such as singular value decomposition and matrix inversion, during objective computation, which can introduce biased gradient estimates and hinder scalability to large systems. In this work, we propose a scalable and conceptually simple method for learning the top-$k$ singular functions of the Koopman operator for stochastic dynamical systems based on the idea of low-rank approximation. Our approach eliminates the need for unstable linear-algebraic operations and integrates easily into modern deep learning pipelines. Empirical results demonstrate that the learned singular subspaces are both reliable and effective for downstream tasks such as eigen-analysis and multi-step prediction.

Deep Learning · Algorithms

Sungje Park, Stephen Tu

Backward stochastic differential equation (BSDE)-based deep learning methods provide an alternative to Physics-Informed Neural Networks (PINNs) for solving high-dimensional partial differential equations (PDEs), offering potential algorithmic advantages in settings such as stochastic optimal control, where the PDEs of interest are tied to an underlying dynamical system. However, standard BSDE-based solvers have empirically been shown to underperform relative to PINNs in the literature. In this paper, we identify the root cause of this performance gap as a discretization bias introduced by the standard Euler-Maruyama (EM) integration scheme applied to one-step self-consistency BSDE losses, which shifts the optimization landscape off target. We find that this bias cannot be satisfactorily addressed through finer step-sizes or multi-step self-consistency losses. To properly handle this issue, we propose a Stratonovich-based BSDE formulation, which we implement with stochastic Heun integration. We show that our proposed approach completely eliminates the bias issues faced by EM integration. Furthermore, our empirical results show that our Heun-based BSDE method consistently outperforms EM-based variants and achieves competitive results with PINNs across multiple high-dimensional benchmarks. Our findings highlight the critical role of integration schemes in BSDE-based PDE solvers, an algorithmic detail that has received little attention thus far in the literature.