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Applications · Physics

Alberto Pepe, Mattia Montanari, Joan Lasenby

We introduce Fengbo, a pipeline entirely in Clifford Algebra to solve 3D partial differential equations (PDEs) specifically for computational fluid dynamics (CFD). Fengbo is an architecture composed of only 3D convolutional and Fourier Neural Operator (FNO) layers, all working in 3D Clifford Algebra. It models the PDE solution problem as an interpretable mapping from the geometry to the physics of the problem. Despite having just few layers, Fengbo achieves competitive accuracy, superior to 5 out of 6 proposed models reported in \cite{li2024geometry} for the $\emph{ShapeNet Car}$ dataset, and it does so with only 42 million trainable parameters, at a reduced computational complexity compared to graph-based methods, and estimating jointly pressure \emph{and} velocity fields. In addition, the output of each layer in Fengbo can be clearly visualised as objects and physical quantities in 3D space, making it a whitebox model. By leveraging Clifford Algebra and establishing a direct mapping from the geometry to the physics of the PDEs, Fengbo provides an efficient, geometry- and physics-aware approach to solving complex PDEs.

Deep Learning · Large Language Models

Zhiyang He, Sashrika Pandey, Mariah Schrum, Anca Dragan

To deliver high-quality, personalized responses, large language models (LLMs) must effectively incorporate context — personal, demographic, and cultural information specific to an end-user. For example, asking the model to explain Newton's second law with the context "I am a toddler'' should produce a response different from when the context is "I am a physics professor''. However, leveraging the context in practice is a nuanced and challenging task, and is often dependent on the specific situation or user base. The model must strike a balance between providing specific, personalized responses and maintaining general applicability. Current solutions, such as prompt-engineering and fine-tuning, require collection of contextually appropriate responses as examples, making them time-consuming and less flexible to use across different contexts. In this work, we introduce Context Steering (CoS) —a simple, training-free decoding approach that amplifies the influence of the context in next token predictions. CoS computes contextual influence by comparing the output probabilities from two LLM forward passes: one that includes the context and one that does not. By linearly scaling the contextual influence, CoS allows practitioners to flexibly control the degree of personalization for different use cases. We show that CoS can be applied to autoregressive LLMs, and demonstrates strong performance in personalized recommendations. Additionally, we show that CoS can function as a Bayesian Generative model to infer and quantify correlations between open-ended texts, broadening its potential applications.

Applications · Genetics, Cell Biology, Health, etc

Sichen Zhu, Yuchen Zhu, Molei Tao, Peng Qiu

Spatial Transcriptomics (ST) allows a high-resolution measurement of RNA sequence abundance by systematically connecting cell morphology depicted in Hematoxylin and eosin (H\&E) stained histology images to spatially resolved gene expressions. ST is a time-consuming, expensive yet powerful experimental technique that provides new opportunities to understand cancer mechanisms at a fine-grained molecular level, which is critical for uncovering new approaches for disease diagnosis and treatments. Here, we present $\textbf{Stem}$ ($\underline{\textbf{S}}$pa$\underline{\textbf{T}}$ially resolved gene $\underline{\textbf{E}}$xpression inference with diffusion $\underline{\textbf{M}}$odel), a novel computational tool that leverages a conditional diffusion generative model to enable in silico gene expression inference from H&E stained images. Through better capturing the inherent stochasticity and heterogeneity in ST data, $\textbf{Stem}$ achieves state-of-the-art performance on spatial gene expression prediction and generates biologically meaningful gene profiles for new H&E stained images at test time. We evaluate the proposed algorithm on datasets with various tissue sources and sequencing platforms, where it demonstrates clear improvement over existing approaches. $\textbf{Stem}$ generates high-fidelity gene expression predictions that share similar gene variation levels as ground truth data, suggesting that our method preserves the underlying biological heterogeneity. Our proposed pipeline opens up the possibility of analyzing existing, easily accessible H&E stained histology images from a genomics point of view without physically performing gene expression profiling and empowers potential biological discovery from H&E stained histology images. Code is available at: https://github.com/SichenZhu/Stem.

Applications · Physics

Youn-Yeol Yu, Jeongwhan Choi, Jaehyeon Park, Kookjin Lee, Noseong Park

Recently, data-driven simulators based on graph neural networks have gained attention in modeling physical systems on unstructured meshes. However, they struggle with long-range dependencies in fluid flows, particularly in refined mesh regions. This challenge, known as the 'over-squashing' problem, hinders information propagation. While existing graph rewiring methods address this issue to some extent, they only consider graph topology, overlooking the underlying physical phenomena. We propose Physics-Informed Ollivier--Ricci Flow (PIORF), a novel rewiring method that combines physical correlations with graph topology. PIORF uses Ollivier--Ricci curvature (ORC) to identify bottleneck regions and connects these areas with nodes in high-velocity gradient nodes, enabling long-range interactions and mitigating over-squashing. Our approach is computationally efficient in rewiring edges and can scale to larger simulations. Experimental results on 3 fluid dynamics benchmark datasets show that PIORF consistently outperforms baseline models and existing rewiring methods, achieving up to 26.2\% improvement.

Applications · Physics

Tuan Truong, Rithwik Sudharsan, Yibo Yang, Peter Xiangyuan, Ruihan Yang, Stephan Mandt, Joshua Bloom

The site conditions that make astronomical observatories in space and on the ground so desirable---cold and dark---demand a physical remoteness that leads to limited data transmission capabilities. Such transmission limitations directly bottleneck the amount of data acquired and in an era of costly modern observatories, any improvements in lossless data compression has the potential scale to billions of dollars worth of additional science that can be accomplished on the same instrument. Traditional lossless methods for compressing astrophysical data are manually designed. Neural data compression, on the other hand, holds the promise of learning compression algorithms end-to-end from data and outperforming classical techniques by leveraging the unique spatial, temporal, and wavelength structures of astronomical images. This paper introduces [AstroCompress](https://huggingface.co/AstroCompress): a neural compression challenge for astrophysics data, featuring four new datasets (and one legacy dataset) with 16-bit unsigned integer imaging data in various modes: space-based, ground-based, multi-wavelength, and time-series imaging. We provide code to easily access the data and benchmark seven lossless compression methods (three neural and four non-neural, including all practical state-of-the-art algorithms).Our results on lossless compression indicate that lossless neural compression techniques can enhance data collection at observatories, and provide guidance on the adoption of neural compression in scientific applications. Though the scope of this paper is restricted to lossless compression, we also comment on the potential exploration of lossy compression methods in future studies.

Applications · Genetics, Cell Biology, Health, etc

Konstantin Hemker, Nikola Simidjievski, Mateja Jamnik

Learning holistic computational representations in physical, chemical or biological systems requires the ability to process information from different distributions and modalities within the same model. Thus, the demand for multimodal machine learning models has sharply risen for modalities that go beyond vision and language, such as sequences, graphs, time series, or tabular data. While there are many available multimodal fusion and alignment approaches, most of them require end-to-end training, scale quadratically with the number of modalities, cannot handle cases of high modality imbalance in the training set, or are highly topology-specific, making them too restrictive for many biomedical learning tasks. This paper presents Multimodal Lego (MM-Lego), a general-purpose fusion framework to turn any set of encoders into a competitive multimodal model with no or minimal fine-tuning. We achieve this by introducing a wrapper for any unimodal encoder that enforces shape consistency between modality representations. It harmonises these representations by learning features in the frequency domain to enable model merging with little signal interference. We show that MM-Lego 1) can be used as a model merging method which achieves competitive performance with end-to-end fusion models without any fine-tuning, 2) can operate on any unimodal encoder, and 3) is a model fusion method that, with minimal fine-tuning, surpasses all benchmarks in five out of seven datasets.

Reinforcement Learning · Batch/Offline

Cevahir Koprulu, Franck Djeumou, ufuk topcu

Offline model-based reinforcement learning (RL) offers a principled approach to using a learned dynamics model as a simulator to optimize a control policy. Despite the near-optimal performance of existing approaches on benchmarks with high-quality datasets, most struggle on datasets with low state-action space coverage or suboptimal demonstrations.We develop a novel offline model-based RL approach that particularly shines in low-quality data regimes while maintaining competitive performance on high-quality datasets.Neural Stochastic Differential Equations for Uncertainty-aware, Offline RL (NUNO) learns a dynamics model as neural stochastic differential equations (SDE), where its drift term can leverage prior physics knowledge as inductive bias.In parallel, its diffusion term provides distance-aware estimates of model uncertainty by matching the dynamics' underlying stochasticity near the training data regime while providing high but bounded estimates beyond it.To address the so-called model exploitation problem in offline model-based RL, NUNO builds on existing studies by penalizing and adaptively truncating neural SDE's rollouts according to uncertainty estimates.Our empirical results in D4RL and NeoRL MuJoCo benchmarks evidence that NUNO outperforms state-of-the-art methods in low-quality datasets by up to 93% while matching or surpassing their performance by up to 55% in some high-quality counterparts.

Applications · Physics

Yi Xiao, Qilong Jia, Kun Chen, LEI BAI, Wei Xue

Data assimilation (DA) is an essential statistical technique for generating accurate estimates of a physical system's states by combining prior model predictions with observational data, especially in the realm of weather forecasting. Effectively modeling the prior distribution while adapting to diverse observational sources presents significant challenges for both traditional and neural network-based DA algorithms. This paper introduces VAE-Var, a novel neural network-based data assimilation algorithm aimed at 1) enhancing accuracy by capturing the non-Gaussian characteristics of the conditional background distribution $p(\mathbf{x}|\mathbf{x}_b)$, and 2) efficiently assimilating real-world observational data. VAE-Var utilizes a variational autoencoder to learn the background error distribution, with its decoder creating a variational cost function to optimize the analysis states. The advantages of VAE-Var include: 1) it maintains the framework of traditional variational assimilation, enabling it to accommodate various observation operators, particularly irregular observations; 2) it lessens the dependence on expert knowledge for constructing the background distribution, allowing for improved modeling of non-Gaussian structures; and 3) experimental results indicate that, when applied to the FengWu weather forecasting model, VAE-Var outperforms DiffDA and two traditional algorithms (interpolation and 3DVar) in terms of assimilation accuracy in sparse observational contexts, and is capable of assimilating real-world GDAS prepbufr observations over a year.

Applications · Physics

Shuhao Cao, Francesco Brarda, Ruipeng Li, Yuanzhe Xi

Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training expenses, and may not always achieve the desired accuracy required in many scientific and engineering disciplines. Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training expenses, and may not always achieve the desired accuracy required in many scientific and engineering disciplines. In this paper, we propose a new learning framework to address these issues. A new spatiotemporal adaptation is proposed to generalize any Fourier Neural Operator (FNO) variant to learn maps between Bochner spaces, which can perform an arbitrary-length temporal super-resolution for the first time. To better exploit this capacity, a new paradigm is proposed to refine the commonly adopted end-to-end neural operator training and evaluations with the help from the wisdom from traditional numerical PDE theory and techniques. Specifically, in the learning problems for the turbulent flow modeled by the Navier-Stokes Equations (NSE), the proposed paradigm trains an FNO only for a few epochs. Then, only the newly proposed spatiotemporal spectral convolution layer is fine-tuned without the frequency truncation. The spectral fine-tuning loss function uses a negative Sobolev norm for the first time in operator learning, defined through a reliable functional-type a posteriori error estimator whose evaluation is exact thanks to the Parseval identity. Moreover, unlike the difficult nonconvex optimization problems in the end-to-end training, this fine-tuning loss is convex. Numerical experiments on commonly used NSE benchmarks demonstrate significant improvements in both computational efficiency and accuracy, compared to end-to-end evaluation and traditional numerical PDE solvers under certain conditions. The source code is publicly available at https://github.com/scaomath/torch-cfd.

Applications · Physics

Chuan Liu, Chunshu Wu, shihui cao, Mingkai Chen, James Liang, Ang Li, Michael Huang, Chuang Ren, Yingnian Wu, Dongfang Liu 等

The rapid development of AI highlights the pressing need for sustainable energy, a critical global challenge for decades. Nuclear fusion, generally seen as a promising solution, has been the focus of intensive research for nearly a century, with investments reaching hundreds of billions of dollars. Recent advancements in Inertial Confinement Fusion (ICF) have drawn significant attention to fusion research, in which Laser-Plasma Interaction (LPI) is critical for ensuring fusion stability and efficiency. However, the complexity of LPI makes analytical approaches impractical, leaving researchers dependent on extremely computationally intensive Particle-in-Cell (PIC) simulations to generate data, posing a significant bottleneck to the advancement of fusion research. In response, this work introduces Diff-PIC, a novel framework that leverages conditional diffusion models as a computationally efficient alternative to PIC simulations for generating high-fidelity scientific LPI data. In this work, physical patterns captured by PIC simulations are distilled into diffusion models associated with two tailored enhancements: (1) To effectively capture the complex relationships between physical parameters and their corresponding outcomes, the parameters are encoded in a physically informed manner. (2) To further enhance efficiency while maintaining physical validity, the rectified flow technique is employed to transform our model into a one-step conditional diffusion model. Experimental results show that Diff-PIC achieves a $\sim$16,200$\times$ speedup compared to traditional PIC on a 100 picosecond simulation, while delivering superior accuracy compared to other data generation approaches.

Deep Learning · Everything Else

Hyunwoo Lee, Hayoung Choi, Hyunju Kim

As a neural network's depth increases, it can improve generalization performance. However, training deep networks is challenging due to gradient and signal propagation issues. To address these challenges, extensive theoretical research and various methods have been introduced. Despite these advances, effective weight initialization methods for tanh neural networks remain insufficiently investigated. This paper presents a novel weight initialization method for neural networks with tanh activation function. Based on an analysis of the fixed points of the function $\tanh(ax)$, the proposed method aims to determine values of $a$ that mitigate activation saturation. A series of experiments on various classification datasets and physics-informed neural networks demonstrates that the proposed method outperforms Xavier initialization methods (with or without normalization) in terms of robustness across different network sizes, data efficiency, and convergence speed. Code is available at https://github.com/1HyunwooLee/Tanh-Init.

Applications · Neuroscience, Cognitive Science

Dehong Xu, Ruiqi Gao, Wenhao Zhang, Xue-Xin Wei, Yingnian Wu

This paper investigates the conformal isometry hypothesis as a potential explanation for the hexagonal periodic patterns in grid cell response maps. We posit that grid cell activities form a high-dimensional vector in neural space, encoding the agent's position in 2D physical space. As the agent moves, this vector rotates within a 2D manifold in the neural space, driven by a recurrent neural network. The conformal hypothesis proposes that this neural manifold is a conformal isometric embedding of 2D physical space, where local physical distance is preserved by the embedding up to a scaling factor (or unit of metric). Such distance-preserving position embedding is indispensable for path planning in navigation, especially planning local straight path segments. We conduct numerical experiments to show that this hypothesis leads to the hexagonal grid firing patterns by learning maximally distance-preserving position embedding, agnostic to the choice of the recurrent neural network. Furthermore, we present a theoretical explanation of why hexagon periodic patterns emerge by minimizing our loss function by showing that hexagon flat torus is maximally distance preserving.

Applications · Language, Speech and Dialog

Mohan Xu, Kai Li, Guo Chen, Xiaolin Hu

In recent years, much speech separation research has focused primarily on improving model performance. However, for low-latency speech processing systems, high efficiency is equally important. Therefore, we propose a speech separation model with significantly reduced parameters and computational costs: Time-frequency Interleaved Gain Extraction and Reconstruction network (TIGER). TIGER leverages prior knowledge to divide frequency bands and compresses frequency information. We employ a multi-scale selective attention module to extract contextual features, while introducing a full-frequency-frame attention module to capture both temporal and frequency contextual information. Additionally, to more realistically evaluate the performance of speech separation models in complex acoustic environments, we introduce a dataset called EchoSet. This dataset includes noise and more realistic reverberation (e.g., considering object occlusions and material properties), with speech from two speakers overlapping at random proportions. Experimental results showed that models trained on EchoSet had better generalization ability than those trained on other datasets to the data collected in the physical world, which validated the practical value of the EchoSet. On EchoSet and real-world data, TIGER significantly reduces the number of parameters by 94.3% and the MACs by 95.3% while achieving performance surpassing state-of-the-art (SOTA) model TF-GridNet.

Reinforcement Learning · Inverse

Anish Abhijit Diwan, Julen Urain, Jens Kober, Jan Peters

This paper introduces a new imitation learning framework based on energy-based generative models capable of learning complex, physics-dependent, robot motion policies through state-only expert motion trajectories. Our algorithm, called Noise-conditioned Energy-based Annealed Rewards (NEAR), constructs several perturbed versions of the expert's motion data distribution and learns smooth, and well-defined representations of the data distribution's energy function using denoising score matching. We propose to use these learnt energy functions as reward functions to learn imitation policies via reinforcement learning. We also present a strategy to gradually switch between the learnt energy functions, ensuring that the learnt rewards are always well-defined in the manifold of policy-generated samples. We evaluate our algorithm on complex humanoid tasks such as locomotion and martial arts and compare it with state-only adversarial imitation learning algorithms like Adversarial Motion Priors (AMP). Our framework sidesteps the optimisation challenges of adversarial imitation learning techniques and produces results comparable to AMP in several quantitative metrics across multiple imitation settings.

Applications · Chemistry and Drug Discovery

Xiaoran Jiao, Weian Mao, Wengong Jin, Peiyuan Yang, Hao Chen, Chunhua Shen

Predicting the change in binding free energy ($\Delta \Delta G$) is crucial for understanding and modulating protein-protein interactions, which are critical in drug design.Due to the scarcity of experimental $\Delta\Delta G$ data, existing methods focus on pre-training, while neglecting the importance of alignment.In this work, we propose Boltzmann Alignment technique to transfer knowledge from pre-trained inverse folding models to prediction of $\Delta\Delta G$.We begin by analyzing the thermodynamic definition of $\Delta\Delta G$ and introducing the Boltzmann distribution to connect energy to the protein conformational distribution. However, the protein conformational distribution is intractable. Therefore, we employ Bayes’ theorem to circumvent direct estimation and instead utilize the log-likelihood provided by protein inverse folding models for the estimation of $\Delta\Delta G$. Compared to previous methods based on inverse folding, our method explicitly accounts for the unbound state of the protein complex in the $\Delta \Delta G$ thermodynamic cycle, introducing a physical inductive bias and achieving supervised and unsupervised state-of-the-art (SoTA) performance.Experimental results on SKEMPI v2 indicate that our method achieves Spearman coefficients of 0.3201 (unsupervised) and 0.5134 (supervised) on SKEMPI v2, significantly surpassing the previously reported %SoTA valuesSoTA results of 0.2632 and 0.4324, respectively.Furthermore, we demonstrate the capability of our method in bindingenergy prediction, protein-protein docking, and antibody optimization tasks.Code is available at [https://github.com/aim-uofa/BA-DDG](https://github.com/aim-uofa/BA-DDG)

Dmitry Kamzolov, Artem Agafonov, Dmitry Pasechnyuk, Alexander Gasnikov, Martin Takáč

Second-order methods for convex optimization outperform first-order methods in terms of theoretical iteration convergence, achieving rates up to $O(k^{-5})$ for highly-smooth functions. However, their practical performance and applications are limited due to their multi-level structure and implementation complexity. In this paper, we present new results on high-order optimization methods, supported by their practical performance. First, we show that the basic high-order methods, such as the Cubic Regularized Newton Method, exhibit global superlinear convergence for $\mu$-strongly star-convex functions, a class that includes $\mu$-strongly convex functions and some non-convex functions. Theoretical convergence results are both inspired and supported by the practical performance of these methods. Secondly, we propose a practical version of the Nesterov Accelerated Tensor method, called NATA. It significantly outperforms the classical variant and other high-order acceleration techniques in practice. The convergence of NATA is also supported by theoretical results. Finally, we introduce an open-source computational library for high-order methods, called OPTAMI. This library includes various methods, acceleration techniques, and subproblem solvers, all implemented as PyTorch optimizers, thereby facilitating the practical application of high-order methods to a wide range of optimization problems. We hope this library will simplify research and practical comparison of methods beyond first-order.

Applications · Language, Speech and Dialog

Muhammad Shah, David Solans Noguero, Mikko Heikkilä, Bhiksha Raj, Nicolas Kourtellis

As Automatic Speech Recognition (ASR) models become ever more pervasive, it is important to ensure that they make reliable predictions under corruptions present in the physical and digital world. We propose Speech Robust Bench (SRB), a comprehensive benchmark for evaluating the robustness of ASR models to diverse corruptions. SRB is composed of 114 input perturbations which simulate an heterogeneous range of corruptions that ASR models may encounter when deployed in the wild. We use SRB to evaluate the robustness of several state-of-the-art ASR models and observe that model size and certain modeling choices such as the use of discrete representations, or self-training appear to be conducive to robustness. We extend this analysis to measure the robustness of ASR models on data from various demographic subgroups, namely English and Spanish speakers, and males and females. Our results revealed noticeable disparities in the model's robustness across subgroups. We believe that SRB will significantly facilitate future research towards robust ASR models, by making it easier to conduct comprehensive and comparable robustness evaluations.

Miscellaneous Aspects of Machine Learning · Unsupervised and Semi-supervised Learning

Namgyu Kang, Jaemin Oh, Youngjoon Hong, Eunbyung Park

The numerical approximation of partial differential equations (PDEs) using neural networks has seen significant advancements through Physics-Informed Neural Networks (PINNs). Despite their straightforward optimization framework and flexibility in implementing various PDEs, PINNs often suffer from limited accuracy due to the spectral bias of Multi-Layer Perceptrons (MLPs), which struggle to effectively learn high-frequency and nonlinear components. Recently, parametric mesh representations in combination with neural networks have been investigated as a promising approach to eliminate the inductive bias of MLPs. However, they usually require high-resolution grids and a large number of collocation points to achieve high accuracy while avoiding overfitting. In addition, the fixed positions of the mesh parameters restrict their flexibility, making accurate approximation of complex PDEs challenging. To overcome these limitations, we propose Physics-Informed Gaussians (PIGs), which combine feature embeddings using Gaussian functions with a lightweight neural network. Our approach uses trainable parameters for the mean and variance of each Gaussian, allowing for dynamic adjustment of their positions and shapes during training. This adaptability enables our model to optimally approximate PDE solutions, unlike models with fixed parameter positions. Furthermore, the proposed approach maintains the same optimization framework used in PINNs, allowing us to benefit from their excellent properties. Experimental results show the competitive performance of our model across various PDEs, demonstrating its potential as a robust tool for solving complex PDEs. Our project page is available at https://namgyukang.github.io/Physics-Informed-Gaussians/

Deep Learning · Sequential Models, Time series

jindong tian, Yuxuan Liang, Ronghui Xu, Peng Chen, Chenjuan Guo, Aoying Zhou, Lujia Pan, Zhongwen Rao, Bin Yang

Air pollution significantly threatens human health and ecosystems, necessitating effective air quality prediction to inform public policy. Traditional approaches are generally categorized into physics-based and data-driven models. Physics-based models usually struggle with high computational demands and closed-system assumptions, while data-driven models may overlook essential physical dynamics, confusing the capturing of spatiotemporal correlations. Although some physics-guided approaches combine the strengths of both models, they often face a mismatch between explicit physical equations and implicit learned representations. To address these challenges, we propose Air-DualODE, a novel physics-guided approach that integrates dual branches of Neural ODEs for air quality prediction. The first branch applies open-system physical equations to capture spatiotemporal dependencies for learning physics dynamics, while the second branch identifies the dependencies not addressed by the first in a fully data-driven way. These dual representations are temporally aligned and fused to enhance prediction accuracy. Our experimental results demonstrate that Air-DualODE achieves state-of-the-art performance in predicting pollutant concentrations across various spatial scales, thereby offering a promising solution for real-world air quality challenges.

Jianxin Zhang, Josh Viktorov, Doosan Jung, Emily Pitler

Neural Stochastic Differential Equations (Neural SDEs) have emerged as powerful mesh-free generative models for continuous stochastic processes, with critical applications in fields such as finance, physics, and biology. Previous state-of-the-art methods have relied on adversarial training, such as GANs, or on minimizing distance measures between processes using signature kernels. However, GANs suffer from issues like instability, mode collapse, and the need for specialized training techniques, while signature kernel-based methods require solving linear PDEs and backpropagating gradients through the solver, whose computational complexity scales quadratically with the discretization steps. In this paper, we identify a novel class of strictly proper scoring rules for comparing continuous Markov processes. This theoretical finding naturally leads to a novel approach called Finite Dimensional Matching (FDM) for training Neural SDEs. Our method leverages the Markov property of SDEs to provide a computationally efficient training objective. This scoring rule allows us to bypass the computational overhead associated with signature kernels and reduces the training complexity from $O(D^2)$ to $O(D)$ per epoch, where $D$ represents the number of discretization steps of the process. We demonstrate that FDM achieves superior performance, consistently outperforming existing methods in terms of both computational efficiency and generative quality.