{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:51:29Z","timestamp":1760147489406,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,8]],"date-time":"2023-02-08T00:00:00Z","timestamp":1675814400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deutsche Forschungsgemeinschft (DFG)","award":["318763901-SFB1294","01IS18025A","01IS18037A"],"award-info":[{"award-number":["318763901-SFB1294","01IS18025A","01IS18037A"]}]},{"name":"BIFOLD-Berlin Institute for the Foundations of Learning and Data","award":["318763901-SFB1294","01IS18025A","01IS18037A"],"award-info":[{"award-number":["318763901-SFB1294","01IS18025A","01IS18037A"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A Schr\u00f6dinger bridge is a stochastic process connecting two given probability distributions over time. It has been recently applied as an approach for generative data modelling. The computational training of such bridges requires the repeated estimation of the drift function for a time-reversed stochastic process using samples generated by the corresponding forward process. We introduce a modified score- function-based method for computing such reverse drifts, which can be efficiently implemented by a feed-forward neural network. We applied our approach to artificial datasets with increasing complexity. Finally, we evaluated its performance on genetic data, where Schr\u00f6dinger bridges can be used to model the time evolution of single-cell RNA measurements.<\/jats:p>","DOI":"10.3390\/e25020316","type":"journal-article","created":{"date-parts":[[2023,2,9]],"date-time":"2023-02-09T01:59:43Z","timestamp":1675907983000},"page":"316","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Score-Based Approach for Training Schr\u00f6dinger Bridges for Data Modelling"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1354-4715","authenticated-orcid":false,"given":"Ludwig","family":"Winkler","sequence":"first","affiliation":[{"name":"Machine Learning Group, Technische Universit\u00e4t Berlin, 10623 Berlin, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4042-8920","authenticated-orcid":false,"given":"Cesar","family":"Ojeda","sequence":"additional","affiliation":[{"name":"Artificial Intelligence Group, Technische Universit\u00e4t Berlin, 10623 Berlin, Germany"},{"name":"Institute of Mathematics, University of Potsdam, 14469 Potsdam, Germany"}]},{"given":"Manfred","family":"Opper","sequence":"additional","affiliation":[{"name":"Artificial Intelligence Group, Technische Universit\u00e4t Berlin, 10623 Berlin, Germany"},{"name":"Institute of Mathematics, University of Potsdam, 14469 Potsdam, Germany"},{"name":"Centre for Systems Modelling and Quantitative Biomedicine, University of Birmingham, Birmingham B15 2TT, UK"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,8]]},"reference":[{"key":"ref_1","unstructured":"Song, Y., Sohl-Dickstein, J., Kingma, D.P., Kumar, A., Ermon, S., and Poole, B. (2020). Score-based generative modeling through stochastic differential equations. arXiv."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Saharia, C., Chan, W., Saxena, S., Li, L., Whang, J., Denton, E., Ghasemipour, S.K.S., Ayan, B.K., Mahdavi, S.S., and Lopes, R.G. (2022). Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding. arXiv.","DOI":"10.1145\/3528233.3530757"},{"key":"ref_3","first-page":"8780","article-title":"Diffusion models beat gans on image synthesis","volume":"34","author":"Dhariwal","year":"2021","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_4","first-page":"6840","article-title":"Denoising diffusion probabilistic models","volume":"33","author":"Ho","year":"2020","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_5","unstructured":"Schr\u00f6dinger, E. (1931). \u00dcber die Umkehrung der Naturgesetze, Verlag der Akademie der Wissenschaften in Kommission bei Walter De Gruyter."},{"key":"ref_6","first-page":"269","article-title":"Sur la th\u00e9orie relativiste de l\u2019\u00e9lectron et l\u2019interpr\u00e9tation de la m\u00e9canique quantique","volume":"2","year":"1932","journal-title":"Ann. L\u2019Inst. Henri Poincar\u00e9"},{"key":"ref_7","unstructured":"L\u00e9onard, C. (2013). A survey of the Schr\u00f6dinger problem and some of its connections with optimal transport. arXiv."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1007\/s10957-015-0803-z","article-title":"On the relation between optimal transport and Schr\u00f6dinger bridges: A stochastic control viewpoint","volume":"169","author":"Chen","year":"2016","journal-title":"J. Optim. Theory Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1017\/S0962492919000011","article-title":"Data assimilation: The Schr\u00f6dinger perspective","volume":"28","author":"Reich","year":"2019","journal-title":"Acta Numer."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1146\/annurev-control-070220-100858","article-title":"Optimal transport in systems and control","volume":"4","author":"Chen","year":"2021","journal-title":"Annu. Rev. Control Robot. Auton. Syst."},{"key":"ref_11","unstructured":"Bernton, E., Heng, J., Doucet, A., and Jacob, P.E. (2019). Schr\u00f6dinger Bridge Samplers. arXiv."},{"key":"ref_12","first-page":"17695","article-title":"Diffusion Schr\u00f6dinger bridge with applications to score-based generative modeling","volume":"34","author":"Thornton","year":"2021","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Vargas, F., Thodoroff, P., Lamacraft, A., and Lawrence, N. (2021). Solving schr\u00f6dinger bridges via maximum likelihood. Entropy, 23.","DOI":"10.3390\/e23091134"},{"key":"ref_14","first-page":"695","article-title":"Estimation of non-normalized statistical models by score matching","volume":"6","author":"Dayan","year":"2005","journal-title":"J. Mach. Learn. Res."},{"key":"ref_15","unstructured":"Oksendal, B. (2013). Stochastic Differential Equations: An Introduction with Applications, Springer Science & Business Media."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"876","DOI":"10.1214\/aoms\/1177703591","article-title":"A relationship between arbitrary positive matrices and doubly stochastic matrices","volume":"35","author":"Sinkhorn","year":"1964","journal-title":"Ann. Math. Stat."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1561\/2200000073","article-title":"Computational optimal transport: With applications to data science","volume":"11","author":"Cuturi","year":"2019","journal-title":"Found. Trends Mach. Learn."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1160","DOI":"10.1214\/aos\/1176324703","article-title":"Convergence of the iterative proportional fitting procedure","volume":"23","author":"Ruschendorf","year":"1995","journal-title":"Ann. Stat."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1103\/PhysRev.150.1079","article-title":"Derivation of the Schr\u00f6dinger equation from Newtonian mechanics","volume":"150","author":"Nelson","year":"1966","journal-title":"Phys. Rev."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/0304-4149(82)90051-5","article-title":"Reverse-time diffusion equation models","volume":"12","author":"Anderson","year":"1982","journal-title":"Stoch. Process. Appl."},{"key":"ref_21","unstructured":"Nelson, E. (1988). \u00c9cole d\u2019\u00c9t\u00e9 de Probabilit\u00e9s de Saint-Flour XV\u2013XVII, 1985\u20131987, Springer."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Maoutsa, D., Reich, S., and Opper, M. (2020). Interacting particle solutions of Fokker\u2013Planck equations through gradient\u2013log\u2013density estimation. Entropy, 22.","DOI":"10.3390\/e22080802"},{"key":"ref_23","unstructured":"Hendrycks, D., and Gimpel, K. (2016). Gaussian error linear units (gelus). arXiv."},{"key":"ref_24","unstructured":"Agarap, A.F. (2018). Deep learning using rectified linear units (relu). arXiv."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1661","DOI":"10.1162\/NECO_a_00142","article-title":"A connection between score matching and denoising autoencoders","volume":"23","author":"Vincent","year":"2011","journal-title":"Neural Comput."},{"key":"ref_26","unstructured":"Boffi, N.M., and Vanden-Eijnden, E. (2022). Probability flow solution of the Fokker\u2013Planck equation. arXiv."},{"key":"ref_27","unstructured":"Ba, J.L., Kiros, J.R., and Hinton, G.E. (2016). Layer normalization. arXiv."},{"key":"ref_28","unstructured":"Kingma, D.P., and Ba, J. (2014). Adam: A method for stochastic optimization. arXiv."},{"key":"ref_29","unstructured":"Loshchilov, I., and Hutter, F. (2016). Sgdr: Stochastic gradient descent with warm restarts. arXiv."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"823","DOI":"10.1103\/PhysRev.36.823","article-title":"On the theory of the Brownian motion","volume":"36","author":"Uhlenbeck","year":"1930","journal-title":"Phys. Rev."},{"key":"ref_31","unstructured":"Glorot, X., and Bengio, Y. (2010, January 13\u201315). Understanding the difficulty of training deep feedforward neural networks. Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, JMLR Workshop and Conference Proceedings, Sardinia, Italy."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"He, K., Zhang, X., Ren, S., and Sun, J. (2015, January 7\u201313). Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. Proceedings of the IEEE International Conference on Computer Vision, Santiago, Chile.","DOI":"10.1109\/ICCV.2015.123"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Villani, C. (2009). Optimal Transport: Old and New, Springer.","DOI":"10.1007\/978-3-540-71050-9"},{"key":"ref_34","unstructured":"Song, Y., and Ermon, S. (2019). Generative modeling by estimating gradients of the data distribution. Adv. Neural Inf. Process. Syst., 11918\u201311930."},{"key":"ref_35","unstructured":"Buitinck, L., Louppe, G., Blondel, M., Pedregosa, F., Mueller, A., Grisel, O., Niculae, V., Prettenhofer, P., Gramfort, A., and Grobler, J. (2013). API design for machine learning software: Experiences from the scikit-learn project. arXiv."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1482","DOI":"10.1038\/s41587-019-0336-3","article-title":"Visualizing structure and transitions in high-dimensional biological data","volume":"37","author":"Moon","year":"2019","journal-title":"Nat. Biotechnol."},{"key":"ref_37","unstructured":"Tong, A., Huang, J., Wolf, G., Van Dijk, D., and Krishnaswamy, S. (2020, January 13\u201318). Trajectorynet: A dynamic optimal transport network for modeling cellular dynamics. Proceedings of the International Conference on Machine Learning, PMLR, Virtual."},{"key":"ref_38","unstructured":"Kloeden, P.E., Platen, E., and Schurz, H. (2002). Numerical Solution of SDE through Computer Experiments, Springer Science & Business Media."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1111\/j.1467-9868.2006.00552.x","article-title":"Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes (with discussion)","volume":"68","author":"Beskos","year":"2006","journal-title":"J. R. Stat. Soc. Ser. B (Stat. Methodol.)"},{"key":"ref_40","first-page":"5","article-title":"On analytic methods in probability theory","volume":"5","author":"Kolmogorov","year":"1938","journal-title":"Uspekhi Mat. Nauk"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1016\/0550-3213(81)90056-0","article-title":"Correlation functions and computer simulations","volume":"180","author":"Parisi","year":"1981","journal-title":"Nucl. Phys. B"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1111\/j.2517-6161.1994.tb02000.x","article-title":"Representations of knowledge in complex systems","volume":"56","author":"Grenander","year":"1994","journal-title":"J. R. Stat. Soc. Ser. B (Methodol.)"},{"key":"ref_43","unstructured":"Welling, M., and Teh, Y.W. (July, January 28). Bayesian learning via stochastic gradient Langevin dynamics. Proceedings of the 28th International Conference on Machine Learning (ICML-11), Washington, DC, USA."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/2\/316\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:28:29Z","timestamp":1760120909000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/2\/316"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,8]]},"references-count":43,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["e25020316"],"URL":"https:\/\/doi.org\/10.3390\/e25020316","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2023,2,8]]}}}