An AI-assisted river model brings physical equations close to deep-learning accuracy

Predicting river flow means tracking more than today’s rain. Water can linger in soil, accumulate as snow, drain through groundwater or wait until wet hillsides become connected to a stream. In a November 2026 preprint, University of Birmingham researcher Xilin Xia asks whether a compact representation of these processes can rival neural networks receiving the same weather and landscape information.

The resulting model, CaRT+, reaches broadly comparable daily streamflow accuracy to individual long short-term memory networks, or LSTMs, in the study’s gauged catchments. That is not an across-the-board victory: neural-network ensembles retain advantages, and predicting catchments excluded from training remains harder.

Physical structure, with memory

Xia organises the model around three principles: conservation, dissipation and aggregation. Conservation requires explicit accounting for water entering, leaving and moving between stores. Dissipation constrains water movement to follow driving potential differences, with phase changes bounded by available water and energy.

Aggregation addresses what happens when a varied landscape is compressed into a few variables. Two catchments—or two internal configurations of one catchment—can hold the same total water yet release different amounts because that water is distributed differently. The paper argues that unresolved differences can enter a compact model through effective flow relationships and memory of previous weather and storage.

For example, hillslope patches may connect to a river only after crossing different wetness thresholds. Averaging those thresholds produces a gradually changing connected fraction rather than one abrupt switch. Additional variables track hillslope wetness, near-surface wetness and vegetation activity, influencing flows without themselves adding water to the balance.

These principles constrain possible equations; they do not uniquely determine them. The paper distinguishes exact aggregation under specified assumptions from approximations, postulated relationships and heuristic choices.

What the AI agents did

The starting model, CaRT, has 11 parameters per catchment. Xia refined it by diagnosing recurring errors, identifying a process hypothesis from the literature, implementing a candidate change and comparing performance across training seeds on development periods.

Claude Code and Codex helped execute this loop under the author’s direction. According to the paper, they shortened the path from a diagnosed deficit to a verified, trained candidate to hours. Xia guided the search and redirected the agents using hydrological understanding; this was not autonomous discovery without human supervision.

CaRT+ has 35 parameters per catchment and 14 principal state variables, alongside additional water held in transit by its routing scheme. Its additions include surface interception, depression storage, vegetation-related evaporation and changing hydrological connectivity.

Crucially, “no daily neural network” does not mean no neural network. A static network maps catchment attributes to parameters, which remain fixed throughout the simulated record. Physical equations then evolve the daily states and flows.

How close were the predictions?

The final evaluation used 1989–2008 for training, 2009–2010 for checkpoint selection and 2011–2019 for testing. Models received daily precipitation, temperature and potential evaporation, plus catchment attributes.

The headline measure is Nash–Sutcliffe efficiency, or NSE, which scores squared prediction errors relative to observed flow variability; higher is better. For single-model comparisons, the study averages each catchment’s score across three training seeds, then reports the median across catchments. This differs from averaging their flow predictions into an ensemble.

Across 602 gauged British catchments, CaRT+ scored 0.863, compared with 0.869 for the stateful LSTM and 0.862 for the entity-aware LSTM. Across 593 US catchments, the corresponding scores were 0.765, 0.751 and 0.754.

Australia provided a test of the model structure beyond its development datasets. The structure was fixed before Australian data were added, but the models were trained on Australian records: this was not zero-shot transfer of British or US parameters. Across 505 gauged Australian catchments, CaRT+ scored 0.778 against 0.762 and 0.753 for the two LSTMs.

Ensembling changes the comparison. Three-seed LSTM ensembles led CaRT+ by 0.003–0.012 in median paired NSE in Britain and the United States. The stateful LSTM ensemble also retained an Australian advantage over the full test period. In dry years in Australia, the single stateful LSTM had the better NSE.

What the results establish—and leave open

Catchments excluded from training exposed another weakness. CaRT+ trailed the entity-aware LSTM on held-out US basins and both LSTMs on held-out Australian catchments. Averaging simulations generated from perturbed attributes reduced some deficits, but missing information about water losses remained a problem.

The paper also discloses that Britain’s snow-store choice was initially adopted after test-period results had been seen, then checked on the development split. The evaluation therefore was not completely untouched by structural decisions.

Finally, accurate streamflow does not independently validate every internal process. Xia calls for soil-moisture, groundwater and evaporation observations to constrain those interpretations. The central finding is narrower: a shared, physically constrained structure can approach individual LSTM accuracy in these gauged datasets. Its proposed universal “laws of form” remain an interpretation supported by the results, not proof that every catchment mechanism has been identified.